The Database of Integer Sequences, Part 14
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
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demo1.html: Demos
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eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A091712
%S A091712 1,2,2,2,3,6,14,36,99,286,858,2652,8398,27132,89148,297160,1002915,
%T A091712 3421710,11785890,40940460,143291610,504932340,1790214660,6382504440,
%U A091712 22870640910,82334307276,297670187844,1080432533656,3935861372604
%N A091712 a(n)=6(2n-4)!/((n-2)!n!), if n>2. a(0)=1,a(1)=a(2)=2.
%F A091712 G.f.: ((1+10x-2x^2)+(1-4x)^(3/2))/2. a(n)=6(2n-4)!/((n-2)!n!), if n>2. a(n)=a(n-1)(4n-10)/n, if n>3.
%F A091712 G.f. A(x) = (2c(x)-1)^3/c(x)^4 = (1-c(x)x)(1+c(x)x)^3, where c(x) = g.f. for Catalan numbers A000108.
%o A091712 (PARI) a(n)=if(n<3,(n>=0)+(n>0),6*(2*n-4)!/n!/(n-2)!)
%o A091712 (PARI) a(n)=if(n<0,0,polcoeff(((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2,n))
%o A091712 (PARI) a(n)=if(n<=0,n==0,polcoeff(subst((1-x)*(1+x)^3,x,serreverse(x-x^2+x*O(x^n))),n))
%Y A091712 A007054(n)=a(n+2), if n>0.
%Y A091712 Essentially the same as A007054.
%Y A091712 Adjacent sequences: A091709 A091710 A091711 this_sequence A091713 A091714 A091715
%Y A091712 Sequence in context: A038715 A057040 A096235 this_sequence A125721 A049798 A024682
%K A091712 nonn
%O A091712 0,2
%A A091712 Michael Somos, Jan 31 2004
%I A125721
%S A125721 2,2,2,3,6,15,48,168,840,4536,26880,147840,1209600
%N A125721 a(n)=2*n!/d(n!); d(m)=A000005(m) is the number of divisors of m.
%C A125721 a(3)=3 and a(5)=15 are the only odd numbers in this sequence.
%D A125721 P. Erdos, solved by J. Fiedler, Elem. Math. 16 (1961), 42-44, Aufgabe 374.
%F A125721 a(n)=2*A000142(n)/A027423(n).
%e A125721 a(4)=2*4!/d(4!)=2*24/8=6.
%Y A125721 Cf. A000142, A027423.
%Y A125721 Adjacent sequences: A125718 A125719 A125720 this_sequence A125722 A125723 A125724
%Y A125721 Sequence in context: A057040 A096235 A091712 this_sequence A049798 A024682 A091228
%K A125721 nonn
%O A125721 0,1
%A A125721 Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Feb 01 2007
%I A049798
%S A049798 0,0,0,1,0,2,2,2,3,7,2,7,10,8,8,15,11,19,16,15,22,32,19,25,34,34,33,46,
%T A049798 33,47,47,48,61,65,45,62,77,79,68,87,74,94,97,86,105,127,98,114,120,
%U A049798 124,129,154,141,151,142,147,172,200,151,180
%N A049798 a(n)=(1/2)*Sum{T(n,k): k=2,3,...,n}, array T as in A049800.
%Y A049798 Adjacent sequences: A049795 A049796 A049797 this_sequence A049799 A049800 A049801
%Y A049798 Sequence in context: A096235 A091712 A125721 this_sequence A024682 A091228 A134890
%K A049798 nonn
%O A049798 1,6
%A A049798 Clark Kimberling (ck6(AT)evansville.edu)
%I A024682
%S A024682 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,7,6,8,7,11,9,18,17,20,25,
%T A024682 25,32,36,38,40,42,49,59,62,69,68,75,91,105,104,114,112,120,134,141,149,160,
%U A024682 172,183,193,195,203,218,255,257,271,268,317,333,342,368,358,379,413,432
%N A024682 a(n) = number of ways p(n) is a sum of 3 odd nonprimes r,s,t satisfying 15 <= r < s < t.
%Y A024682 Adjacent sequences: A024679 A024680 A024681 this_sequence A024683 A024684 A024685
%Y A024682 Sequence in context: A091712 A125721 A049798 this_sequence A091228 A134890 A101360
%K A024682 nonn
%O A024682 1,21
%A A024682 Clark Kimberling (ck6(AT)evansville.edu)
%I A091228
%S A091228 2,2,2,3,7,7,7,7,11,11,11,11,13,13,19,19,19,19,19,19,25,25,25,25,25,25,
%T A091228 31,31,31,31,31,31,37,37,37,37,37,37,41,41,41,41,47,47,47,47,47,47,55,
%U A091228 55,55,55,55,55,55,55,59,59,59,59,61,61,67,67,67,67,67,67,73,73,73
%N A091228 Smallest m >= n, such that m is irreducible when interpreted as GF(2)[X]-polynomial.
%C A091228 Analogous to A007918.
%H A091228 A. Karttunen, Scheme-program for computing this sequence.
%H A091228 Index entries for sequences operating on GF(2)[X]-polynomials
%F A091228 a(n) = n + A091229(n).
%Y A091228 Adjacent sequences: A091225 A091226 A091227 this_sequence A091229 A091230 A091231
%Y A091228 Sequence in context: A125721 A049798 A024682 this_sequence A134890 A101360 A110910
%K A091228 nonn
%O A091228 0,1
%A A091228 Antti Karttunen (His-Firstname.His-Surname(AT)iki.fi), Jan 03 2004
%I A134890
%S A134890 2,2,2,3,7,16,15,7,4,5,12,28,33,17,8,7,13,35,52,31,13,9,14,37,68,50,21,
%T A134890 11,14,36,78,74,33,15,15,32,80,99,51,21,16,29,75,120,77,30,18,26,67,132,
%U A134890 108,45,22,24,57,132,141,66,28
%N A134890 Ceiling(n*exp(cos n)).
%Y A134890 Adjacent sequences: A134887 A134888 A134889 this_sequence A134891 A134892 A134893
%Y A134890 Sequence in context: A049798 A024682 A091228 this_sequence A101360 A110910 A119532
%K A134890 nonn
%O A134890 1,1
%A A134890 Mohammad K. Azarian (azarian(AT)evansville.edu), Nov 14 2007
%I A101360
%S A101360 2,2,2,3,7,22,83,326,1300,5195,20776,83098,332387,1329543,5318166,
%T A101360 21272659,85090631,340362521,1361450080,5445800316,21783201259,
%U A101360 87132805033,348531220128,1394124880509,5576499522030,22305998088117
%N A101360 Ceiling((3-sqrt(3))*4^(n-3)) + 1
%C A101360 An approximation to the Camel Problem.
%p A101360 Digits:=100;seq(ceil((3-sqrt(3))*4^(n-3)) + 1,n=0..30);
%Y A101360 Cf. A094062.
%Y A101360 Adjacent sequences: A101357 A101358 A101359 this_sequence A101361 A101362 A101363
%Y A101360 Sequence in context: A024682 A091228 A134890 this_sequence A110910 A119532 A010583
%K A101360 nonn
%O A101360 0,1
%A A101360 Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 25 2004
%I A110910
%S A110910 1,2,2,2,3,8,13,15,49,22,17,17,16,26,29,41,34,25,21,26,21,21,36,31,29,
%T A110910 95,25,29,34,38,105,150,61,582,43,58,92,108,263,277,50,212,59,53,57,99,
%U A110910 55,170,196,812,105,54,53,85,59,81,0,418,63,63,314,117,118,170,236,104
%N A110910 Configurations in the evolution of a line of n cells in Conway's Game of Life, with 0=infinity. For periodic evolutions, a(n)=(preperiod length)+(period length). For non-periodic evolutions, a(n)=0.
%C A110910 If nothing catches up with an outbound glider, then a(n)=0 for n>=1000 because when you watch the horizontal 1000-line evolve in a simulator, around the 490th generation, gliders fly away from the left and right corners _before_ the non-chaotic growing in the middle has finished, so you will see the same local picture in the 490th generation of longer lines.
%D A110910 Berlekamp/Conway/Guy, Winning Ways ..., 2nd ed, vol. 4, chapter 25
%e A110910 a(0)=1 because there is only the empty configuration. a(10)=2+15 because the 10-line needs two steps to become a pentadecathlon. a(56)=0 because the 56-line sends four gliders to outer space.
%o A110910 {- Haskell program for verification of periodic cases. The
%o A110910 non-periodic cases listed here evolve into a periodic kernel plus
%o A110910 gliders whose paths ahead do not intersect each other or the kernel
%o A110910 (gliders marching in single file are not counted as intersecting).
%o A110910 Replace leading dots by spaces before running! -}
%o A110910 import Data.Set
%o A110910 main = print [if n `elem` known then 0 else a n | n<-[0..105]]
%o A110910 known = [56,71,72,75,78,82,85,86,87,88,91,92,93,94,96,98,100,102,103,105]
%o A110910 a n = count empty (iterate evolve (fromList [(x,0) | x<-[1..n]]))
%o A110910 neighbors (x,y) = fromList
%o A110910 ................. [(x+u,y+v) | u<-[ -1,0,1], v<-[ -1,0,1], (u,v)/=(0,0)]
%o A110910 evolve life =
%o A110910 . let fil f = Data.Set.filter
%o A110910 ............. (\x-> f (size (life `intersection` neighbors x)))
%o A110910 . in (life `difference` fil (\k-> k<2 || k>3) life) `union` fil (== 3)
%o A110910 .... (unions (Prelude.map neighbors (elems life)) `difference` life)
%o A110910 count o (x:xs) | x `member` o = 0
%o A110910 .............. | otherwise = 1 + count (o `union` singleton x) xs
%Y A110910 Cf. A061342 A019473 A056605 A056614 A055397 A099733 A089520 A098720 A056613.
%Y A110910 Adjacent sequences: A110907 A110908 A110909 this_sequence A110911 A110912 A110913
%Y A110910 Sequence in context: A091228 A134890 A101360 this_sequence A119532 A010583 A051007
%K A110910 nonn,uned
%O A110910 0,2
%A A110910 Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 03 2005
%I A119532
%S A119532 2,2,2,3,8,16,47,132,435,1445,5103,18259,66934,247826,928137,3500559,
%T A119532 13290552,50712016,194358380,747624825,2885120934,11164896416,
%U A119532 43313276948,168400053019,656028153120,2560227845391,10007858038797
%N A119532 Number of n-ominoes plus number of n-iamonds.
%C A119532 Number of polyominoes with n cells plus number of polyiamonds with n cells. Turning over is allowed; holes are allowed.
%F A119532 a(n) = A000105(n) + A000577(n).
%Y A119532 Cf. A000105, A000577.
%Y A119532 Adjacent sequences: A119529 A119530 A119531 this_sequence A119533 A119534 A119535
%Y A119532 Sequence in context: A134890 A101360 A110910 this_sequence A010583 A051007 A071470
%K A119532 nonn
%O A119532 0,1
%A A119532 Jonathan Vos Post (jvospost2(AT)yahoo.com), May 28 2006
%I A010583
%S A010583 2,2,2,3,9,8,0,0,9,0,5,6,9,3,1,5,5,2,1,1,6,5,3,6,3,3,7,6,7,2,2,1,5,
%T A010583 7,1,9,6,5,1,8,6,9,9,1,2,8,0,9,6,9,2,3,0,5,5,6,9,9,3,4,5,8,0,8,6,6,
%U A010583 0,4,0,0,9,8,3,0,8,2,9,7,5,9,7,4,4,8,9,7,5,8,0,5,4,4,8,1,6,2,6,2,7
%N A010583 Decimal expansion of cube root of 11.
%Y A010583 Adjacent sequences: A010580 A010581 A010582 this_sequence A010584 A010585 A010586
%Y A010583 Sequence in context: A101360 A110910 A119532 this_sequence A051007 A071470 A104461
%K A010583 nonn,cons
%O A010583 1,1
%A A010583 njas
%I A051007
%S A051007 0,2,2,2,3,12,131,1,7,1,2,1,3,3,1,2,5,39,2,1,169,2,2,2,1,1,2,5,1,2,1,
%T A051007 199,24,7,7,1,163,1,3,2,1,2,14,1,3,5,1,1,1,1,3,3,2,1,279,1,4,1,1,4,1,92,
%U A051007 1,16,2,3,1,5,2,25,9,1,1,2,8,1,2,5,1,1,3,1,1,2,1,7,1,1,1,5,1,2,2,5,6,1
%N A051007 Continued fraction for prime constant A051006.
%H A051007 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%H A051007 G. Xiao, Contfrac
%H A051007 Index entries for continued fractions for constants
%t A051007 ContinuedFraction[ FromDigits[{{Table[ If[ PrimeQ[n], 1, 0], {n, 370}]}, 0}, 2], 95] (from Robert G. Wilson v Jan 15 2005)
%Y A051007 Cf. A010051, A051006. Increasing partial quotients are in A102878.
%Y A051007 Adjacent sequences: A051004 A051005 A051006 this_sequence A051008 A051009 A051010
%Y A051007 Sequence in context: A110910 A119532 A010583 this_sequence A071470 A104461 A084862
%K A051007 nonn,cofr
%O A051007 0,2
%A A051007 Eric Weisstein (eric(AT)weisstein.com)
%I A071470
%S A071470 1,0,0,2,2,2,4,1,1,1,3,3,2,4,4,4,6,6,6,2,3,1,1,1,7,6,6,8,4,2,1,1,7,
%T A071470 6,5,3,4,8,1,1
%N A071470 Sprague-Grundy values for octal game .146y.
%D A071470 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071470 Adjacent sequences: A071467 A071468 A071469 this_sequence A071471 A071472 A071473
%Y A071470 Sequence in context: A119532 A010583 A051007 this_sequence A104461 A084862 A027387
%K A071470 nonn
%O A071470 1,4
%A A071470 njas and Sue Pope (pope(AT)research.att.com), May 29 2002
%I A104461
%S A104461 0,1,1,2,2,2,4,1,5,3,2,5,4,1,7,4,2,3,4,5,4,4,2,5,7,1,5,8,4,4,8,1,10,2,4,
%T A104461 5,5,3,5,7,4,2,14,1,7,5,8,4,5,4,5,12,2,9,4,4,5,11,4,2,13,8,1,5,7,8,5,4,
%U A104461 4,1,5,13,2,7,9,5,8,14,2,10,5,5,10,4,5,5,8,1,5,23,2,2,5,4,6,7,6,4,8,13
%N A104461 Number of instances of nonprimes m in pythagorean triples x,y,z such that x^2+y^2=z^2. Except for 1, the number of instances of composite numbers m in pythagorean triples.
%C A104461 The Pari script is direct and very fast for m = x,y values but slows in the trial routine for m=z. We save some for m even allowing to test only even values of y.
%F A104461 Consider pythagorean triples x^2+y^2=z^2. We seek to find the total number of instances of an integer m being x or y or z. The solution for x or y is straight forward by considering appropriate lesser and greater pairwise factors, L, G of m^2 in z^2 - y^2 = (z-y)(z+y) = m^2. Then solve for z and y with the relations, z-y = L z+y = G 2z = L+G, z = (L+G)/2 where L and G are both even if m is even or both odd if m is odd. The number of L factors < m is the number of instances of x or y. The count of instances z=m is solved by trial on x^2 = m^2 - y^2.
%e A104461 For m=30 there are 5 pythagorean triples that have a 30:
%e A104461 30,224,226
%e A104461 30,72,78
%e A104461 30,40,50
%e A104461 30,16,34
%e A104461 18,24,30
%o A104461 (PARI) for(k=1,400,if(isprime(k)==0,print1(pythm3(k)","))) \instances of m in pythagorean triples using a direct method for x,y pythm3(m) = { local(m2,ln,j,j2=0,d,d2,q2,q,a,b,x,x1,x2,xx,y,y2,z,c,c2,r,f,str,stp); d=divisors(m^2); \get the divisors of m^2. ln=length(d)-1; d2=q2=vector(ln); m2=m^2; if(m%2,r=1,r=0); for(j=1,ln, \save only the both even r=0, both odd r=1 if(d[j]%2==r, if(m2/d[j]%2==r, j2++; d2[j2]=d[j]; q2[j2]=m2/d[j]; \save m/factor to solve (z-y)(z+y) = m^2 ) ) ); x2=y2 = vector(20); for(j=1,j2, z=(d2[j] + q2[j])/2; y= z - d2[j]; if(y>0, c++; ) ); if(m%2==0,start=2;step=2,start=1;step=1); forstep(y=start,m-1,step, esolve when z is m x1 = (m2-y^2); if(issquare(x1), c2++; x2[c2]=floor(sqrt(x1)); \save to later mask dupes y2[c2]=y; ) ); for(x=1,c2, \mask the dupes routine for(y=x,c2, if(x2[x]==y2[y], ) ) ); return(c+c2/2) \print total }
%Y A104461 Cf. A046081 A088978.
%Y A104461 Adjacent sequences: A104458 A104459 A104460 this_sequence A104462 A104463 A104464
%Y A104461 Sequence in context: A010583 A051007 A071470 this_sequence A084862 A027387 A111735
%K A104461 easy,nonn
%O A104461 1,4
%A A104461 Cino Hilliard (hillcino368(AT)gmail.com), Apr 18 2005
%I A084862
%S A084862 2,2,2,4,1,34,4,2,3,5,1,33,2,3,1,1,12,1,20,1,9,1,2,1,4,3,1,13,1,1,2,3,1,
%T A084862 1,8,6,1,1,3,42,1,94,1,4,2,1,1,1,7,1,1,1,16,1,25,2,1,1,29,2,3,2,3,10,6,
%U A084862 2,1,1,1,2,1,4,2,132,1,3,1,2,8,1,1,2,1,3,1,2,1,3,1,3,45,1,2,1,5
%N A084862 Continued fraction expansion of sum( zeta( 2n ) / n!, n = 1 .. infinity ).
%H A084862 Eric Weisstein's World of Mathematics, Riemann Zeta Function.
%e A084862 2.40744655479032851470948665622302272558226649037984418869339833...
%Y A084862 Decimal expansion is in A076813.
%Y A084862 Adjacent sequences: A084859 A084860 A084861 this_sequence A084863 A084864 A084865
%Y A084862 Sequence in context: A051007 A071470 A104461 this_sequence A027387 A111735 A102298
%K A084862 nonn,easy,cofr
%O A084862 1,1
%A A084862 Frank.Ellermann(AT)t-online.de, Jul 13 2003
%I A027387
%S A027387 0,2,2,2,4,2,2,2,0,1,2,4,4,4,6,4,3,4,2,3,2,4,4,4,6,4,3
%N A027387 Write digits for n, count endpoints (version 2).
%Y A027387 Adjacent sequences: A027384 A027385 A027386 this_sequence A027388 A027389 A027390
%Y A027387 Sequence in context: A071470 A104461 A084862 this_sequence A111735 A102298 A049298
%K A027387 nonn
%O A027387 0,2
%A A027387 njas
%I A111735
%S A111735 1,2,2,2,4,2,2,2,2,2,4,2,4,2,8,4,2,8,10,10,4,2,2,2,2,4,2,10,4,8,2,4,8,2,
%T A111735 2,4,8,2,2,2,4,4,4,8,2,2,8,4,2,4,2,2,4,4,2,8,2,8,8,10,4,2,8,4,2,2,4,2,8,
%U A111735 2,2,10,2,4,14,2,4,2,10,2,2,14,4,2,2,32,14,2,16,10,8,2,10,8,2,2,4,4,2,4
%N A111735 Distance between k*(n-th prime) and next prime, k=3 case.
%C A111735 Other cases: k=1 A001223 Differences between consecutive primes, k=2 A059787, k=4 A111736, k=5 A111737, k=6 A111738, k=7 A111739, k=8 A111740, k=9 A111741, k=10 A111742.
%Y A111735 Cf. A001223, A059787, A111736, A111737. A111738, A111739, A111740, A111741, A111742.
%Y A111735 Adjacent sequences: A111732 A111733 A111734 this_sequence A111736 A111737 A111738
%Y A111735 Sequence in context: A104461 A084862 A027387 this_sequence A102298 A049298 A075016
%K A111735 nonn
%O A111735 1,2
%A A111735 Zak Seidov (zakseidov(AT)yahoo.com), Nov 18 2005
%I A102298
%S A102298 2,2,2,4,2,2,2,2,3,3,2,2,2,4,2,4,3,2,2,3,2,3,2,4,2,2,3,6,2,3,2,3,3,3,2,
%T A102298 3,4,2,2,2,2,4,2,3,2,2,2,6,3,4,3,2,2,5,2,3,3,2,2,5,2,2,2,3,3,4,2,3,2,2,
%U A102298 4,4,2,2,2,6,2,2,3,3,3,3,2,4,2,6,2,5,3,2,2,3,3,3,3,5,2,2,2,4,2,3,2,3,4
%N A102298 Number of prime divisors with multiplicity of n+1 where n and n+1 are composite or twin composite numbers.
%e A102298 For n=8 n+1 = 9 = 3*3 or 2 prime divisors with multiplicity.
%o A102298 (PARI) f(n) = for(x=1,n,y=composite(x)+1;if(!isprime(y),print1(bigomega(y)","))) composite(n) =\The n-th composite number. 1 is def as not prime nor composite. { local(c,x); c=1; x=1; while(c <= n, x++; if(!isprime(x),c++); ); return(x) }
%Y A102298 Adjacent sequences: A102295 A102296 A102297 this_sequence A102299 A102300 A102301
%Y A102298 Sequence in context: A084862 A027387 A111735 this_sequence A049298 A075016 A102445
%K A102298 easy,nonn
%O A102298 1,1
%A A102298 Cino Hilliard (hillcino368(AT)gmail.com), Feb 19 2005
%I A049298
%S A049298 0,0,2,2,2,4,2,2,2,4,2,4,2,4,3,2,2,4,2,4,3,4,2,4,2,4,2,4,2,6,2,2,3,4,3,
%T A049298 4,2,4,3,4,2,6,2,4,3,4,2,4,2,4,3,4,2,4,3,4,3,4,2,6,2,4,3,2,3,6,2,4,3,6,
%U A049298 2,4,2,4,3,4,3,6,2,4,2,4,2,6,3,4,3,4,2,6,3,4,3,4,3,4,2,4,3,4,2,6,2,4,5
%N A049298 Take reduced residue systems of n, generate its first differences, dRRS(n); sequence gives maximal value of dRSSS(n).
%C A049298 Greatest values occur at primorial numbers (A002110).
%e A049298 If n is prime, its reduced residue system consists of all numbers below n. But the difference 2 arises from d=1-(n-1)=-n+2 (mod n).
%Y A049298 Cf. A048670. Essentially same as A048669.
%Y A049298 Adjacent sequences: A049295 A049296 A049297 this_sequence A049299 A049300 A049301
%Y A049298 Sequence in context: A027387 A111735 A102298 this_sequence A075016 A102445 A027389
%K A049298 nonn
%O A049298 1,3
%A A049298 Labos E. (labos(AT)ana.sote.hu)
%I A075016
%S A075016 2,2,2,4,2,2,2,4,4,2,12,4,105,2,2,4,7,4,18,22,2,12,11,4,27,118,4,106,21,
%T A075016 2,23,14,12,34,2,4,112,18,105,22,15,2,39,34,7,14,9,4,141,52,7,118,58,4,
%U A075016 12,106,18,50,38,22,10,54,106,14,157
%N A075016 Smallest k such that the concatenation k, k-1,k-2 is divisible by n; or 0 if no such number exists.
%e A075016 a(11) = 12 as 11 divides 121110.
%Y A075016 Cf. A075113, A075114, A075115, A075117.
%Y A075016 Adjacent sequences: A075013 A075014 A075015 this_sequence A075017 A075018 A075019
%Y A075016 Sequence in context: A111735 A102298 A049298 this_sequence A102445 A027389 A049716
%K A075016 base,nonn
%O A075016 1,1
%A A075016 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 01 2002
%E A075016 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
%I A102445
%S A102445 0,1,1,1,2,2,2,4,2,2,3,2,2,4,3,4,6,3,2,3,3,5,6,6,4,9,3,3,2,3,3,4,5,3,5,
%T A102445 4,2,3,3,4,2,7,5,7,7,5,5,6,6,4,5,8,9,4,5,6,3,3,7,6,8,7,7,4,5,4,4,7,7,9,
%U A102445 11,5,8,7,7,6,7,7,8,12,4,7,6,6,4,8,7,4,10,7,7,6,6,7,5,5,6,8,7,9,10,5,7
%N A102445 Number of prime divisors (with multiplicity) of the central trinomial coefficients (A002426).
%C A102445 Prime central trinomial coefficients by index: 2,3,4,(2000); semiprimes by index: 5,6,7,9,10,12,13,19,29,37,41,108,(125); 3 prime factors by index: 11,15,18,20,21,27,28,30,31,34,38,39,57,58,105,(125)
%C A102445 First occurrence of k: 1,2,5,11,8,22,17,42,52,26,89,71,80,....
%H A102445 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics..
%t A102445 bigomega[n_Integer] := Plus @@ Last /@ FactorInteger[n]; tn[n_] := Sum[Binomial[n, k]*Binomial[n - k, k], {k, 0, n/2}]; Table[bigomega[tn[n]], {n, 103}] (from Robert G. Wilson v Feb 21 2005)
%Y A102445 Cf. A002426.
%Y A102445 Adjacent sequences: A102442 A102443 A102444 this_sequence A102446 A102447 A102448
%Y A102445 Sequence in context: A102298 A049298 A075016 this_sequence A027389 A049716 A066671
%K A102445 nonn
%O A102445 1,5
%A A102445 Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 21 2005
%E A102445 Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 21 2005
%I A027389
%S A027389 0,2,2,2,4,2,2,4,0,1,2,4,4,4,6,4,3,6,2,3,2,4,4,4,6,4,3
%N A027389 Write decimal digits for n, count endpoints (version 4).
%Y A027389 Adjacent sequences: A027386 A027387 A027388 this_sequence A027390 A027391 A027392
%Y A027389 Sequence in context: A049298 A075016 A102445 this_sequence A049716 A066671 A049627
%K A027389 nonn,base
%O A027389 0,2
%A A027389 njas
%I A049716
%S A049716 1,2,2,2,4,2,2,4,2,2,4,2,4,6,2,2,4,6,2,4,2,2,4,2,4,6,2,4,6,2,2,
%T A049716 4,6,2,4,2,2,4,6,2,4,2,4,6,2,4,6,8,2,4,2,2,4,2,2,4,2,4,6,8,10,12,
%U A049716 14,2,4,2,4,6,2,2,4,6,8,10,2,2,4,6,2,4,6,2,4,2,4,6,2,4,6,2,2,4
%N A049716 2*n+1-prevprime(2*n+1).
%e A049716 n...: 1 2 3 4 .5 .6 .7 .8 ...
%e A049716 2n+1: 3 5 7 9 11 13 15 17 ...
%e A049716 pp..: 2 3 5 7 .7 11 13 13 ...
%e A049716 diff: 1 2 2 2 .4 .2 .2 .4 ...
%Y A049716 Cf. A049613, A049653, A049711, A049847.
%Y A049716 Adjacent sequences: A049713 A049714 A049715 this_sequence A049717 A049718 A049719
%Y A049716 Sequence in context: A075016 A102445 A027389 this_sequence A066671 A049627 A134058
%K A049716 nonn
%O A049716 1,2
%A A049716 njas
%I A066671
%S A066671 2,2,2,4,2,2,4,2,2,4,4,4,4,4,8,4,8,8,4,4,8,2,2,4,8,4,8,8,4,2,16,4,4,8,
%T A066671 8,8,8,8,2,8,8,8,8,8,4,2,32,8,16,16,4,2,8,16,16,8,8,2,32,16,16,8,8,4,
%U A066671 16,4,16,16,4,8,32,16,8,16,16,2,2,16,4,8,16,4,8,2,16,8,32,4,64,32,32
%N A066671 Powers of 2 arising in A066669: a(n) is the largest even divisor of EulerPhi[A066669(n)], which is by definition is a power of 2.
%e A066671 First, 4th and 15th terms in A066669 are 7, 13, 35; Phi[7] = 2.3, Phi[13] = 4.3, Phi[35] = 24 = 8.3; the largest even divisors[powers of 2] are 2, 4, 8; so a(1) = 2, a(4) = 4, a(15) = 8.
%Y A066671 Cf. A000010, A066669-A066673, A065966.
%Y A066671 Adjacent sequences: A066668 A066669 A066670 this_sequence A066672 A066673 A066674
%Y A066671 Sequence in context: A102445 A027389 A049716 this_sequence A049627 A134058 A086973
%K A066671 nonn
%O A066671 1,1
%A A066671 Labos E. (labos(AT)ana.sote.hu), Dec 18 2001
%I A049627
%S A049627 1,2,2,2,4,2,2,5,5,2,2,6,6,6,2,2,7,8,8,7,2,2,8,9,10,9,8,2,2,9,11,12,12,
%T A049627 11,9,2,2,10,12,15,14,15,12,10,2,2,11,14,16,18,18,16,14,11,2,2,12,15,
%U A049627 19,19,22,19,19,15,12,2,2,13,17,21,23,24,24,23,21
%N A049627 Array T read by diagonals; T(i,j)=(i+1)*(j+1)-H(i,j), where H is the array in A049615; thus T(i,j) is the number of lattice points in rectangle having diagonal (0,0)-to-(i,j) that are visible from (i,j).
%e A049627 Diagonals (each starting on row 1): {1}; {2,2}; {2,4,2}; ...
%Y A049627 Adjacent sequences: A049624 A049625 A049626 this_sequence A049628 A049629 A049630
%Y A049627 Sequence in context: A027389 A049716 A066671 this_sequence A134058 A086973 A029658
%K A049627 nonn,tabl
%O A049627 0,2
%A A049627 Clark Kimberling (ck6(AT)evansville.edu)
%I A134058
%S A134058 1,2,2,2,4,2,2,6,6,2,2,8,12,8,2,2,10,20,20,10,2,2,12,30,40,30,12,2,2,14,
%T A134058 42,70,70,42,14,2,2,16,56,112,140,112,56,16,2,2,18,72,168,252,252,168,
%U A134058 72,18,2
%N A134058 Triangle read by rows, T(n,k) = 2*binomial(n,k) if k>0, (0<=k<=n), left column = (1,2,2,2,...).
%C A134058 Row sums = A046055: (1, 4, 8, 16, 32, 64,...). A134059 = analogous triangle, replacing (1,2,2,2,...) with (1,3,3,3,...).
%C A134058 Triangle T(n,k), 0<=k<=n, read by rows given by [2, -1, 0, 0, 0, 0, 0, ...]DELTA [2, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 07 2007
%F A134058 Double Pascal's triangle and replace leftmost column with (1,2,2,2,...). M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.
%e A134058 First few rows of the triangle are:
%e A134058 1
%e A134058 2, 2;
%e A134058 2, 4, 2;
%e A134058 2, 6, 6, 2;
%e A134058 2, 8, 12, 8, 2;
%e A134058 2, 10, 20, 20, 10, 2;
%e A134058 ...
%Y A134058 Cf. A046055, A134059.
%Y A134058 Adjacent sequences: A134055 A134056 A134057 this_sequence A134059 A134060 A134061
%Y A134058 Sequence in context: A049716 A066671 A049627 this_sequence A086973 A029658 A086327
%K A134058 nonn,tabl
%O A134058 0,2
%A A134058 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 05 2007
%I A086973
%S A086973 0,0,0,0,0,0,2,2,2,4,2,2,6,6,8,8,14,14,2,2,28,28,6,6,62,62,16,16,126,
%T A086973 126,28,28,112,60,6,6,10,30,2,56,2640,2,4,6
%N A086973 Period of oscillator reached starting with a segment of n consecutive live cells and applying the LongLife 2D rule (see comment).
%C A086973 LongLife rule : birth occurs if dead cell has 3,4 or 5 neighbors. Survival occurs if live cell has 5 neighbors. (Sequence computed using LifeLab for mac).
%H A086973 E. Weisstein, Game of Life from mathworld.
%Y A086973 Adjacent sequences: A086970 A086971 A086972 this_sequence A086974 A086975 A086976
%Y A086973 Sequence in context: A066671 A049627 A134058 this_sequence A029658 A086327 A069930
%K A086973 nonn
%O A086973 1,7
%A A086973 Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 22 2003
%I A029658
%S A029658 2,2,2,4,2,2,16,14,6,2,30,20,2,36,50,8,2,2,64,140,196,182,112,44,10,2,
%T A029658 204,336,378,294,156,54,2,100,540,714,672,450,210,12,2,1254,1386,1122,
%U A029658 660,2,144,506,1210,2640,2508,1782,352,90,14,2,650,1716,5148,4290,442
%N A029658 Even numbers in the (2,1)-Pascal triangle A029653.
%Y A029658 Adjacent sequences: A029655 A029656 A029657 this_sequence A029659 A029660 A029661
%Y A029658 Sequence in context: A049627 A134058 A086973 this_sequence A086327 A069930 A066761
%K A029658 nonn,tabf
%O A029658 0,1
%A A029658 Mohammad K. Azarian (ma3(AT)evansville.edu)
%E A029658 More terms from James A. Sellers (sellersj(AT)math.psu.edu)
%I A086327
%S A086327 1,2,2,2,4,2,3,4,4,2,5,2,4,6,4,2,7,2
%N A086327 Number of factors over Q in the factorization of the Chebyshev polynomial of the second kind U_n(x).
%Y A086327 Cf. A001227.
%Y A086327 Adjacent sequences: A086324 A086325 A086326 this_sequence A086328 A086329 A086330
%Y A086327 Sequence in context: A134058 A086973 A029658 this_sequence A069930 A066761 A108920
%K A086327 nonn
%O A086327 1,2
%A A086327 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003
%I A069930
%S A069930 0,1,2,2,2,4,2,3,4,4,2,6,2,4,6,4,2,7,2,6,6,4,2,8,4,4,6,6,2,10,2,5,6,4,
%T A069930 6,10,2,4,6,8,2,10,2,6,10,4,2,10,4,7,6,6,2,10,6,8,6,4,2,14,2,4,10,6,6,
%U A069930 10,2,6,6,10,2,13,2,4,10,6,6,10,2,10,8,4,2,14,6,4,6,8,2,16,6,6,6,4,6
%N A069930 Number of integers of the form (n+k)/(n-k) with 1<=k<=n-1.
%F A069930 a(n) = A032741(n) + A069283(n) = A000005(n) - 1 + A001227(n) - 1 = tau(n) + A001227(n) - 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
%F A069930 Asymptotic formula: since sum(k=1, n, a(k)) = sum(k=1, n, tau(k)) + sum(k=1, n, A001227(k)) - 2*n = A006218(n) + A060831(n) - 2*n = 2*A006218(n) - A006218(floor(n/2)) - 2*n with A006218(0) = 0, A006218(n) = sum(k=1, n, tau(k)) and now, by Dirichlet's asymptotic expression A006218(n) = n*ln(n) + n*(2*gamma-1) + O(n^theta) (gamma = 0.57721..; 1/4 <= theta < 1/2), we have sum(k=1, n, a(k)) = 2*n*ln(n) - (n/2)*ln(n) + o(n*ln(n)) = 1.5*n*ln(n) + o(n*ln(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
%o A069930 (PARI) for(n=1,150,print1(sum(i=1,n-1,if((n+i)%(n-i),0,1)),","))
%Y A069930 Cf. A032741, A069283, A000005, A001227, A006218, A060831, A066743.
%Y A069930 A066660(n) - 1.
%Y A069930 Adjacent sequences: A069927 A069928 A069929 this_sequence A069931 A069932 A069933
%Y A069930 Sequence in context: A086973 A029658 A086327 this_sequence A066761 A108920 A079405
%K A069930 easy,nonn
%O A069930 1,3
%A A069930 Benoit Cloitre (benoit7848c(AT)orange.fr), May 05 2002
%I A066761
%S A066761 1,2,2,2,4,2,3,4,5,2,7,2,5,7,4,2,8,2,7,8,5,2,10,4,5,6,7,2,15,2,5,8,5,7,
%T A066761 13,2,5,8,10,2,15,2,8,12,5,2,13,4,9,8,8,2,12,8,10,8,5,2,23,2,5,13,6,8,
%U A066761 15,2,8,8,16,2,17,2,5,13,8,7,16,2,13,8,5,2,23,8,5,8,10,2,26,7,8,8,5,8
%N A066761 Number of positive integers of the form (n^2+k^2)/(n-k) for k=1,2,3,4,....,n-1.
%C A066761 Also the number of factors of 2*n^2 which are less than n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 12 2002
%C A066761 Also the number of factors of 2*n^2 which are greater than 2*n, so a(n) = tau(2*n^2)-1-A055081(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 13 2002
%F A066761 No general formula is known but let k be a positive integer, p and q distinct odd primes then a(2^k)=k a(p^k)=2*k a(p*q)= 7 or 8 if p >13 a(2*p)= 5 if p>5 a(9*p^2)= 23 .... Asymptotic formula : (1/n)*sum(i=1, n, a(i))= ln(n)*ln(ln(n))+o(ln(n))
%e A066761 a(2)=1 because (2^2+1)/(2-1) is the only integer of this form.
%Y A066761 Adjacent sequences: A066758 A066759 A066760 this_sequence A066762 A066763 A066764
%Y A066761 Sequence in context: A029658 A086327 A069930 this_sequence A108920 A079405 A072048
%K A066761 nonn
%O A066761 2,2
%A A066761 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 17 2002
%E A066761 Corrected by Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 12 2002
%I A108920
%S A108920 0,1,2,2,2,4,2,3,4,5,2,7,2,5,7,4,2,8,2,7,8,5,2,10,4,5,6,7,2,15,2,5,8,5,
%T A108920 7,13,2,5,8,10,2,15,2,8,12,5,2,13,4,9,8,8,2,12,8,10,8,5,2,23,2,5,13,6,8,
%U A108920 15,2,8,8,16,2,17,2,5,13,8,7,16,2,13,8,5,2,23,8,5,8,10,2,26,7,8,8,5,8
%N A108920 Number of positive integers k>n such that n+k divides n^2+k^2.
%C A108920 If n+k divides n^2+k^2 then k<=n(2n+1). If n>2 then there are at least two values of k>n such that n+k divides n^2+k^2; they are k=n(n-1) and k=n(2n-1). Further, if n is prime, these are the only two values. If n=2^j, then there are exactly j values of k>x such that n+k divides n^2+k^2; they are k=3n, k=7n, k=15n,..., k=(2x-1)n. Is this sequence the same as A066761 except for the prepended a(1)=0?
%e A108920 6+k divides 36+k^2 only for k=12,18,30,and 66, so a(6)=4.
%Y A108920 Cf. A066761.
%Y A108920 Adjacent sequences: A108917 A108918 A108919 this_sequence A108921 A108922 A108923
%Y A108920 Sequence in context: A086327 A069930 A066761 this_sequence A079405 A072048 A072056
%K A108920 nonn
%O A108920 1,3
%A A108920 John W. Layman (layman(AT)math.vt.edu), Jul 19 2005
%I A079405
%S A079405 2,2,2,4,2,3,5,3,4,4,3,6,4,5,7,4,4,4,7,5,6,6,5,5,6,2,3,5,3,4,7,4,7,5,6,
%T A079405 4,7,6,8,7,7,5,4,5,7,5,4,6,8,6,6,6,6,5,8,7,7,7,10,6,7,6
%N A079405 Number of dots in primes in Braille.
%H A079405 American Foundation for the Blind, Braille Bug
%e A079405 The 5-th prime is 11, hence a(11)=1+1=2
%o A079405 (PARI) { braille=[3,1,2,2,3,2,3,4,3,2]; forprime (n=2,300, b=braille[n%10+1]; n2=n; if (n>99, b=b+braille[n\100+1]; n2=n%100); if (n2>9,b=b+braille[n2\10+1]); print1(b",")) }
%Y A079405 Cf. A079399, A072283.
%Y A079405 Adjacent sequences: A079402 A079403 A079404 this_sequence A079406 A079407 A079408
%Y A079405 Sequence in context: A069930 A066761 A108920 this_sequence A072048 A072056 A066012
%K A079405 nonn
%O A079405 0,1
%A A079405 Jon Perry (perry(AT)globalnet.co.uk), Feb 16 2003
%I A072048
%S A072048 1,2,2,2,4,2,4,2,2,4,4,2,2,4,4,2,4,2,8,2,4,4,4,2,4,4,2,8,2,4,2,4,2,4,4,
%T A072048 4,2,2,4,4,8,2,4,8,2,2,4,4,8,2,4,2,4,4,4,2,4,4,4,4,2,2,8,2,8,4,2,2,8,4,
%U A072048 2,8,4,4,4,4,4,2,4,8,2,4,4,2,8,2,4,4,4,4,4,2,2,8,4,2
%N A072048 Number of divisors of the square-free numbers: tau(A005117(n)).
%C A072048 a(n) = 2^A072047(n) = 2^A001221(A005117(n)).
%C A072048 Also the number of cube-free numbers with the same square-free kernel as the n-th square-free number, see A073245.
%Y A072048 a(n) = A000005(A005117(n)), A062822.
%Y A072048 Adjacent sequences: A072045 A072046 A072047 this_sequence A072049 A072050 A072051
%Y A072048 Sequence in context: A066761 A108920 A079405 this_sequence A072056 A066012 A063375
%K A072048 nonn
%O A072048 1,2
%A A072048 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 09 2002
%I A072056
%S A072056 2,2,2,4,2,4,4,4,2,2,6,6,2,4,4,2,4,4,8,4,6,4,2,2,8,4,6,4,4,2,8,2,6,6,4,
%T A072056 4,12,4,4,2,2,6,2,6,4,8,6,4,8,8,2,2,8,2,4,4,6,4,8,2,10,2,8,4,8,4,8,12,
%U A072056 4,4,4,2,12,6,8,4,4,8,4,12,2,4,2,6,4,2,4,8,4,6,8,4,12
%N A072056 Number of divisors of 2*prime(n)+1.
%e A072056 Divisors of A072055(8)=2*A000040(8)+1=2*19+1=39: {1,3,13,39} with size 4, therefore a(8)=4.
%Y A072056 a(n)=A000005(A072055(n)), A000040, A072057.
%Y A072056 Adjacent sequences: A072053 A072054 A072055 this_sequence A072057 A072058 A072059
%Y A072056 Sequence in context: A108920 A079405 A072048 this_sequence A066012 A063375 A064129
%K A072056 nonn
%O A072056 1,1
%A A072056 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 11 2002
%I A066012
%S A066012 2,2,2,4,2,4,4,4,2,4,4,4,4,6,6,8,6,8,6,8,8,8,10,10
%N A066012 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z which does not have all Euclidean norms divisible by 8, that is, is strictly Type I. Compare A105681.
%H A066012 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%H A066012 S. T. Dougherty, M. Harada and P. Sole', Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
%H A066012 P. Gaborit, Tables of Self-Dual Codes
%H A066012 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).
%Y A066012 Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.
%Y A066012 Cf. A066013 for number of codes. See also A066014-A066017.
%Y A066012 Adjacent sequences: A066009 A066010 A066011 this_sequence A066013 A066014 A066015
%Y A066012 Sequence in context: A079405 A072048 A072056 this_sequence A063375 A064129 A005137
%K A066012 nonn
%O A066012 1,1
%A A066012 njas, Dec 11 2001
%I A063375
%S A063375 1,1,2,2,2,4,2,4,4,4,2,15,2,4,8,8,2,16,4,16,8,4,2,72,6,4,16,16,2,64,4,
%T A063375 16,8,4,8,160,8,8,8,64,4,64,2,32,32,8,2,336,8,48,8,16,4,128,16,96,32,8,
%U A063375 4,960,4,8,32,64,8,64,8,32,32,128,4,1536,4,16,48,32,16,128,4,512,128,8
%N A063375 Number of divisors of Fibonacci(n).
%o A063375 (PARI) j=[]; for(n=1,150,j=concat(j,numdiv(fibonacci(n)))); j
%Y A063375 Adjacent sequences: A063372 A063373 A063374 this_sequence A063376 A063377 A063378
%Y A063375 Sequence in context: A072048 A072056 A066012 this_sequence A064129 A005137 A105681
%K A063375 easy,nonn
%O A063375 1,3
%A A063375 Jason Earls (jcearls(AT)cableone.net), Jul 23 2001
%I A064129
%S A064129 2,2,2,4,2,4,4,4,4,2,4,2,4,4,8,8,4,4,2,2,2,2,8,4,4,8,4,4,16,8,16,4,8,4,
%T A064129 32,4,4,8,8,16,8,8,16,16,16,8
%N A064129 Number of divisors of 12^n - 1 that are relatively prime to 12^m - 1 for all 0 < m < n.
%H A064129 Sam Wagstaff, Cunningham Project, Factorizations of 12^n-1, n odd, n<240
%t A064129 a = {1}; Do[ d = Divisors[ 12^n - 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 12^n - 1 ] ][ [ 1 ] ] ] ] ], {n, 1, 46} ]
%Y A064129 Cf. A063982.
%Y A064129 Adjacent sequences: A064126 A064127 A064128 this_sequence A064130 A064131 A064132
%Y A064129 Sequence in context: A072056 A066012 A063375 this_sequence A005137 A105681 A130127
%K A064129 nonn
%O A064129 1,1
%A A064129 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2001
%I A005137 M0233
%S A005137 0,2,2,2,4,2,4,4,4,4,4,6,8,6,6,6,8,6,8,8,8,8,8,10,12,10,10,10,12,10,12
%N A005137 Highest minimal distance of self-dual code of length 2n.
%D A005137 J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1334.
%D A005137 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. xxiv.
%Y A005137 Adjacent sequences: A005134 A005135 A005136 this_sequence A005138 A005139 A005140
%Y A005137 Sequence in context: A066012 A063375 A064129 this_sequence A105681 A130127 A098069
%K A005137 nonn,hard
%O A005137 0,2
%A A005137 njas
%I A105681
%S A105681 2,2,2,4,2,4,4,6,2,4,4,4,4,6,6,8,6,8,6,8,8,8,10,12
%N A105681 Highest minimal Lee distance of any Type 4^Z self-dual code of length n over Z/4Z.
%D A105681 W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
%H A105681 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%H A105681 S. T. Dougherty, M. Harada and P. Sole', Shadow Codes over Z_4, Finite Fields Applic., 7 (2001), 507-529.
%H A105681 P. Gaborit, Tables of Self-Dual Codes
%H A105681 E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).
%Y A105681 Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105682.
%Y A105681 See A105688 for the number of such codes. Cf. also A066012.
%Y A105681 Adjacent sequences: A105678 A105679 A105680 this_sequence A105682 A105683 A105684
%Y A105681 Sequence in context: A063375 A064129 A005137 this_sequence A130127 A098069 A135838
%K A105681 nonn
%O A105681 1,1
%A A105681 njas, May 06 2005
%I A130127
%S A130127 1,1,2,2,2,4,2,4,4,8,3,4,8,8,16,3,6,8,16,16,32,4,6,12,16,32,32,64,4,8,
%T A130127 12,24,32,64,64,128,5,8,16,24,48,64,128,128,256,5,10,16,32,48,96,128,
%U A130127 256,256,512
%N A130127 A000012 * A130125.
%C A130127 Row sums = A011377: (1, 3, 8, 18, 39,...). A130126 = A130125 * A000012.
%F A130127 A000012 * A130125
%e A130127 First few rows of the triangle are:
%e A130127 1;
%e A130127 1, 2;
%e A130127 2, 2, 4;
%e A130127 2, 4, 4, 8;
%e A130127 3, 4, 8, 8, 16;
%e A130127 3, 6, 8, 16, 16, 32;
%e A130127 4, 6, 12, 16, 32, 32, 64;
%e A130127 ...
%Y A130127 Cf. A000012, A130125, A130126, A011377.
%Y A130127 Adjacent sequences: A130124 A130125 A130126 this_sequence A130128 A130129 A130130
%Y A130127 Sequence in context: A064129 A005137 A105681 this_sequence A098069 A135838 A114349
%K A130127 nonn,tabl
%O A130127 1,3
%A A130127 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 11 2007
%I A098069
%S A098069 2,2,2,4,2,4,6,8,2,4,4,4,4,6,6,8,8,8,6,8,8,8,10,12
%N A098069 If C is a self-dual code over Z_4 of length N=24m+n, 1 <= n <= 24, then the minimum Lee weight of C is bounded above by 8m+a(n).
%D A098069 E. M. Rains, Bounds for self-dual codes over Z_4, Finite Fields Appl. 6 (2000), 146-163.
%Y A098069 Adjacent sequences: A098066 A098067 A098068 this_sequence A098070 A098071 A098072
%Y A098069 Sequence in context: A005137 A105681 A130127 this_sequence A135838 A114349 A086087
%K A098069 nonn,fini,full
%O A098069 1,1
%A A098069 njas, Sep 13 2004
%I A135838
%S A135838 1,2,2,2,4,2,4,12,12,4,4,16,24,16,4,8,40,80,80,40,8,8,48,120,160,120,48,
%T A135838 8,16,112,336,560,560,336,112,16,16,128,448,896,1120,896,448,128,16,32,
%U A135838 288,1152,2688,4032,4032,2688,1152,288,32
%N A135838 M * A007318, M = a triangle with (1, 2, 2, 4, 4, 8, 8, 16, 16,...) in the main diagonal and the rest zeros.
%C A135838 Row sums = A094015: (1, 4, 8, 32, 64, 256,...)
%F A135838 M * Pascal's triangle as infinite lower triangular matrices, where M = a triangle with (1, 2, 2, 4, 4, 8, 8, 16, 16,...) in the main diagonal and the rest zeros.
%e A135838 First few rows of the triangle are:
%e A135838 1;
%e A135838 2, 2;
%e A135838 2, 4, 2;
%e A135838 4, 12, 12, 4;
%e A135838 4, 16, 24, 16, 4;
%e A135838 8, 40, 80, 80, 40, 8;
%e A135838 ...
%Y A135838 Cf. A094015, A135837.
%Y A135838 Adjacent sequences: A135835 A135836 A135837 this_sequence A135839 A135840 A135841
%Y A135838 Sequence in context: A105681 A130127 A098069 this_sequence A114349 A086087 A133265
%K A135838 nonn,tabl
%O A135838 1,2
%A A135838 Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 01 2007
%I A114349
%S A114349 2,2,2,4,2,6,2,2,6,2,2,6,2,3,2,12,2,3,24,2,2,2,2,9,6,2,3,3,2,3,2,6,
%T A114349 2,30,5,2,2,4,2,36,3,64,2,18,2,2,2,12,8,3,48,3,2,2,3,54,2,2,6,3,2,3,
%U A114349 24,2,2,2,2,12,27,2,2,7,8,2,2,2,2,4,3,60,2,144,4,26,2,2,2,42,2,2,2
%N A114349 Terms of A114331 divided by the appropriate prime (q) in A052248.
%Y A114349 Adjacent sequences: A114346 A114347 A114348 this_sequence A114350 A114351 A114352
%Y A114349 Sequence in context: A130127 A098069 A135838 this_sequence A086087 A133265 A054712
%K A114349 nonn
%O A114349 2,1
%A A114349 njas, based on correspondence from Leroy Quet and Hugo Pfoertner, Feb 22 2006
%I A086087
%S A086087 2,2,2,4,2,6,2,6,4,5,4,4,6
%N A086087 a(n) is the minimal m such that the group GL(m,3) has an element of order n.
%Y A086087 Cf. A085430, A053290.
%Y A086087 Adjacent sequences: A086084 A086085 A086086 this_sequence A086088 A086089 A086090
%Y A086087 Sequence in context: A098069 A135838 A114349 this_sequence A133265 A054712 A066243
%K A086087 nonn
%O A086087 2,1
%A A086087 Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 24 2003
%I A133265
%S A133265 2,2,2,4,2,6,2,8,2,10,2,12,2,14,2,16,2,18,2,20,2,22,2,24,2,26,2,28,2,30,
%T A133265 2,32,2,34,2,36,2,38,2,40,2,42,2,44,2,46,2,48,2,50,2,52,2,54,2,56,2,58,
%U A133265 2,60,2,62,2,64,2,66,2,68,2,70,2,72,2,74,2,76,2,78,2,80
%N A133265 Diagonal of A135356 triangle.
%F A133265 2*(A057979 without 1, 0, first two terms).
%Y A133265 Adjacent sequences: A133262 A133263 A133264 this_sequence A133266 A133267 A133268
%Y A133265 Sequence in context: A135838 A114349 A086087 this_sequence A054712 A066243 A035580
%K A133265 nonn
%O A133265 0,1
%A A133265 Paul Curtz (bpcrtz(AT)free.fr), Dec 20 2007
%I A054712
%S A054712 1,2,2,2,4,2,6,3,3,5,1,2,2,7,5,3,16,3,6,5,7,2,11,3,20,3,4,7,4,5,30,4,2,
%T A054712 17,12,3,9,7,3,6,40,7,42,2,6,12,23,3,42,21,17,3,52,4,4,8,7,5,29,5,15,
%U A054712 31,8,4,4,2,66,17,12,13,35,3,36,10,21,7,6,3,26,6,5,41,41,7,16,43,5,3,8
%N A054712 Number of powers of 12 modulo n.
%Y A054712 Adjacent sequences: A054709 A054710 A054711 this_sequence A054713 A054714 A054715
%Y A054712 Sequence in context: A114349 A086087 A133265 this_sequence A066243 A035580 A135293
%K A054712 easy,nonn
%O A054712 1,2
%A A054712 Henry Bottomley (se16(AT)btinternet.com), Apr 20 2000
%I A066243
%S A066243 2,2,2,4,2,6,4,6,5,10,3,12,7,6,8,16,9,18,4,12,11,22,6,20,13,18,12,28,5
%N A066243 Superseded by A063428.
%Y A066243 Adjacent sequences: A066240 A066241 A066242 this_sequence A066244 A066245 A066246
%Y A066243 Sequence in context: A086087 A133265 A054712 this_sequence A035580 A135293 A053204
%K A066243 dead
%O A066243 2,1
%I A035580
%S A035580 1,0,1,0,1,1,1,2,2,2,4,2,6,6,7,10,8,14,19,16,29,21,37,47,43,70,59,87,116,
%T A035580 101,169,144,206,261,242,374,348,455,581,541,814,779,988,1232,1190,1692,
%U A035580 1700,2056,2565,2505,3453,3545,4196,5169,5160,6841,7214,8319,10219,10304
%N A035580 Number of partitions of n with equal number of parts congruent to each of 1, 3 and 4 (mod 5)
%Y A035580 Adjacent sequences: A035577 A035578 A035579 this_sequence A035581 A035582 A035583
%Y A035580 Sequence in context: A133265 A054712 A066243 this_sequence A135293 A053204 A064025
%K A035580 nonn
%O A035580 0,8
%A A035580 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A135293
%S A135293 2,2,2,4,2,6,10,2,6,18,28,2,6,18,54,82,2,6,18,54,162,244,2,6,18,54,162,
%T A135293 486,730,2,6,18,54,162,486,1458,2188,2,6,18,54,162,486,1458,4374,6562,2,
%U A135293 6,18,54,162,486,1458,4374,13122
%N A135293 Differences between successive numbers n where the sum of digits of n is 2.
%C A135293 The sum of the first N terms, when converted to base 3, gives the Nth term of A052216.
%F A135293 A triangle, where T(0,0)=2, T(n+1,0) = T(n,0)+T(n,n), T(n+1,m) = T(n,m), and T(n+1,n+1) = sum of T(n+1,0..n)
%e A135293 triangle begins:
%e A135293 2
%e A135293 2 2
%e A135293 4 2 6
%e A135293 10 2 6 18
%Y A135293 Cf. A052216.
%Y A135293 Adjacent sequences: A135290 A135291 A135292 this_sequence A135294 A135295 A135296
%Y A135293 Sequence in context: A054712 A066243 A035580 this_sequence A053204 A064025 A054709
%K A135293 nonn,tabl,uned
%O A135293 0,1
%A A135293 Adam Shelly (adam.shelly(AT)gmail.com), Dec 04 2007, Dec 05 2007
%I A053204
%S A053204 1,2,2,2,4,2,10,2,16,8,14,2,28,2,46,38,64,2,46,2,76,50,70,2,136,32,82,
%T A053204 80,156,2,244,2,256,74,38,88,172,2,118,86,256,2,442,2,324,332,326,2,
%U A053204 592,128,274,416,432,2,676,98,648,122,410,2,796,2,934,386,960,292,526
%N A053204 Row sums of A053200.
%C A053204 a(p(n)) = 2, where p(n) = prime numbers A000040, excluding the case where a(1) = 2.
%e A053204 a(6) = 1 + 0 + 3 + 2 + 3 + 0 + 1 = 10
%Y A053204 Cf. A053200.
%Y A053204 Adjacent sequences: A053201 A053202 A053203 this_sequence A053205 A053206 A053207
%Y A053204 Sequence in context: A066243 A035580 A135293 this_sequence A064025 A054709 A121806
%K A053204 nonn,easy
%O A053204 0,2
%A A053204 Asher Auel (asher.auel(AT)reed.edu) Dec 12, 1999
%E A053204 Corrected and extended by James A. Sellers (sellersj(AT)math.psu.edu), Dec 13 1999
%I A064025
%S A064025 1,2,2,2,4,2,16,48,8,4,56,180,44,156,300,7936,10388,11516,9104,
%T A064025 13469268,2684084,2418800,28468692
%N A064025 Length of period of continued fraction for square root of n!.
%e A064025 Quotients for 10! are [[1904], [1, 15, 1, 13, 1, 15, 1, 3808]], so period length of 10! is 8.
%p A064025 with(numtheory): [seq(nops(cfrac(sqrt(k!),'periodic','quotients')[2]),k=2..16)];
%t A064025 Do[ Print[ Length[ Last[ ContinuedFraction[ Sqrt[ n! ]]]]], {n, 2, 24} ]
%Y A064025 Adjacent sequences: A064022 A064023 A064024 this_sequence A064026 A064027 A064028
%Y A064025 Sequence in context: A035580 A135293 A053204 this_sequence A054709 A121806 A056944
%K A064025 cofr,nonn
%O A064025 2,2
%A A064025 Labos E. (labos(AT)ana.sote.hu), Sep 18 2001
%E A064025 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 01 2001
%I A054709
%S A054709 1,2,2,2,4,3,1,2,2,5,10,3,4,2,4,3,8,3,6,5,2,11,11,3,20,5,6,2,28,5,5,3,
%T A054709 10,9,4,3,12,7,4,5,20,3,14,11,4,12,23,4,7,21,8,5,52,7,20,2,6,29,58,5,
%U A054709 20,6,2,3,4,11,22,9,22,5,35,3,3,13,20,7,10,5,13,6,18,21,82,3,8,15,28
%N A054709 Number of powers of 8 modulo n.
%Y A054709 Adjacent sequences: A054706 A054707 A054708 this_sequence A054710 A054711 A054712
%Y A054709 Sequence in context: A135293 A053204 A064025 this_sequence A121806 A056944 A050493
%K A054709 easy,nonn
%O A054709 1,2
%A A054709 Henry Bottomley (se16(AT)btinternet.com), Apr 20 2000
%I A121806
%S A121806 2,2,2,4,3,2,4,2,1,3,4,1,1,1,1,2,2,3,4,2,2,2,3,2,4,1,4,2,1,1,1,1,3,2,4,
%T A121806 3,1,2,2,2,2,1,2,2,4,1,1,4,3,1,4,3,4,2,3,2,1,1,4,3,4,1,1,3,1,3,2,2,4,1,
%U A121806 1,1,1,1,1,1,1,1,1,4,1,4,3,1,2,2,1,1,3,1,1,4,3,1,1,1,4,3,4,2
%N A121806 Primes modulo three as two color partition maps { red, blue} of which there are four types:1-> {red, blue},2->{blue,red},3-> {red,red},4->{blue,blue}.
%C A121806 There are long runs of "1"'s.
%F A121806 a(n) = {1 + Mod[Prime[2*n-1], 3],1 + Mod[Prime[2*n], 3]/. {2, 3} -> 1 /. {3, 2} -> 2 /. { 2, 2} -> 3 /. {3, 3} -> 4
%t A121806 a = Partition[Table[1 + Mod[Prime[n], 3], {n, 3, 203}], 2] /. {2, 3} -> 1 /. {3, 2} -> 2 /. { 2, 2} -> 3 /. {3, 3} -> 4
%Y A121806 Adjacent sequences: A121803 A121804 A121805 this_sequence A121807 A121808 A121809
%Y A121806 Sequence in context: A053204 A064025 A054709 this_sequence A056944 A050493 A085454
%K A121806 nonn,uned
%O A121806 1,1
%A A121806 Roger Bagula (rlbagulatftn(AT)yahoo.com), Aug 29 2006
%I A056944
%S A056944 0,1,2,2,2,4,3,2,4,6,4,2,4,6,8,5,2,4,6,8,10,6,2,4,6,8,10,12,7,2,4,6,8,
%T A056944 10,12,14,8,2,4,6,8,10,12,14,16,9,2,4,6,8,10,12,14,16,18,10,2,4,6,8,10,
%U A056944 12,14,16,18,20,11,2,4,6,8,10,12,14,16,18,20,22,12,2,4,6,8,10,12,14,16
%N A056944 Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.
%C A056944 m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.
%F A056944 a(n) =2n-floor[(sqrt(8n+1)-1)/2]*ceiling[(sqrt(8n+1)-1)/2] =2n-A002024(n)*A003056(n) =2n-A056942(n) =n-A056943(n). If n=t(t+1)/2 then a(n)=t; if n=t(t+1)/2+k with 0Table of n, a(n) for n=0..1024
%F A050493 a(n)=sum([c(n)/b^(i-1)]-[c(n)/b^i]*b, i=1..[log_b(c(n))]+1), b=2, n >= 1, a(0)=0, c(n)=A000217(n)
%Y A050493 A000217, A004157.
%Y A050493 Adjacent sequences: A050490 A050491 A050492 this_sequence A050494 A050495 A050496
%Y A050493 Sequence in context: A054709 A121806 A056944 this_sequence A085454 A083403 A114091
%K A050493 base,easy,nice,nonn
%O A050493 0,3
%A A050493 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999
%I A085454
%S A085454 0,1,2,2,2,4,3,3,6,6,4,5,8,10,8,5,8,11,16,14,10,6,12,16,24,24,18,12,7,17,24,35,40,32,22,14,
%T A085454 8,23,36,51,64,56,40,26,16,9,30,53,75,99,96,72,48,30,18,10,38,76,111,150,160,128,88,56,
%U A085454 34,20,11,47,106,164,225,259,224,160,104,64,38,22
%N A085454 Array defined by T(i,1)=i, T(1,j)=2j, T(i,j)=T(i-1,j)+T(i-1,j-1) read by antidiagonals
%C A085454 Comment from Michael Forbes, (mforbes(AT)alum.MIT.EDU), Sep 16 2005: It seems to me that the recurrance should read: T(i,0)=i, T(0,j)=2j, T(i,j)=T(i-1,j)+T(i-1,j-1) rather than T(i,1)=i, T(1,j)=2j, T(i,j)=T(i-1,j)+T(i-1,j-1). Same for the related sequences, otherwise, is T(1,1) = 1 or 2?
%Y A085454 Cf. A036289 (main diagonal)
%Y A085454 Adjacent sequences: A085451 A085452 A085453 this_sequence A085455 A085456 A085457
%Y A085454 Sequence in context: A121806 A056944 A050493 this_sequence A083403 A114091 A070867
%K A085454 nonn,tabl
%O A085454 1,3
%A A085454 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 12 2003
%I A083403
%S A083403 1,2,2,2,4,3,3,6,6,5,7,8,8,8,10,8,10,11,12,13,15,11,14,14,16,14,15,19,
%T A083403 19,19,18,20,19,21,21,22,24,21,24,26,22,27,26,26,27,27,29,30,30,31,32,
%U A083403 32,33,32,34,35,35,33,36,34,36,37,40,37,42,40,41,43,42,44,40,45,44,46
%N A083403 Write the numbers 1, 2, ... in a triangle with n terms in the n-th row; a(n) = number of square-free integers in n-th row.
%e A083403 Triangle begins
%e A083403 1 (1 squarefree)
%e A083403 2 3 (2 squarefree)
%e A083403 4 5 6 (2 squarefree)
%e A083403 7 8 9 10 (2 squarefree)
%e A083403 11 12 13 14 15 (4 squarefree)
%e A083403 16 17 18 19 20 21 (3 squarefree)
%o A083403 (PARI) {ts(m)=local(r, t); r=1; for(n=1,m,t=0; for(k=r,n+r-1,if(issquarefree(k),t++)); print1(t","); r=n+r;) }
%Y A083403 Cf. A066888, A005117.
%Y A083403 Adjacent sequences: A083400 A083401 A083402 this_sequence A083404 A083405 A083406
%Y A083403 Sequence in context: A056944 A050493 A085454 this_sequence A114091 A070867 A029157
%K A083403 easy,nonn
%O A083403 1,2
%A A083403 Jason Earls (jcearls(AT)cableone.net), Jun 07 2003
%I A114091
%S A114091 1,1,2,2,2,4,3,3,7,4,4,11,5,5,16,6,6,22,7,7,29,8,8,37,9,9,46,10,10,56,
%T A114091 11,11,67,12,12,79,13,13,92,14,14,106,15,15,121,16,16,137,17,17
%N A114091 Number of partitions of n into parts that are distinct mod 3.
%e A114091 a(5)=2 because there are 2 such partition of 5: {5}, {2,3}.
%t A114091 << DiscreteMath`Combinatorica`; np[n_]:= Length@Select[Mod[ #,3]& /@ Partitions[n],(Length@# != Length@Union@#)&]; lst = Array[np,50]
%Y A114091 Adjacent sequences: A114088 A114089 A114090 this_sequence A114092 A114093 A114094
%Y A114091 Sequence in context: A050493 A085454 A083403 this_sequence A070867 A029157 A031437
%K A114091 nonn
%O A114091 1,3
%A A114091 Giovanni Resta (g.resta(AT)iit.cnr.it), Feb 06 2006
%I A070867
%S A070867 1,1,2,2,2,4,3,4,4,4,8,5,5,8,8,6,8,12,8,11,9,9,10,13,16,9,12,20,10,12,
%T A070867 23,12,15,21,13,17,18,19,19,22,21,19,19,26,28,16,24,33,21,26,23,36,16,
%U A070867 26,39,16,30,33,36,19,34,31,43,23,30,32,38,23,40,26,38,43,31,31,38,44
%N A070867 a(1) = a(2) = 1; a(n) = a(n-1 - a(n-1)) + a(n-1 - a(n-2)).
%D A070867 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.
%H A070867 Eric Weisstein's World of Mathematics, Wolfram Sequences
%H A070867 Index entries for Hofstadter-type sequences
%H A070867 Nick Hobson, Python program for this sequence
%t A070867 a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1 - a[n - 1]] + a[n - 1 - a[n - 2]]; Table[ a[n], {n, 1, 80}]
%Y A070867 Adjacent sequences: A070864 A070865 A070866 this_sequence A070868 A070869 A070870
%Y A070867 Sequence in context: A085454 A083403 A114091 this_sequence A029157 A031437 A029145
%K A070867 nonn,easy
%O A070867 1,3
%A A070867 njas, May 19 2002
%E A070867 More terms from John W. Layman (layman(AT)math.vt.edu), May 21 2002
%E A070867 Erroneous comma in sequence corrected by Nick Hobson, Feb 17 2007
%I A029157
%S A029157 1,0,1,1,1,1,2,2,2,4,3,4,5,5,6,7,8,8,11,10,12,14,14,16,
%T A029157 18,19,20,24,24,26,30,30,33,36,38,40,45,46,49,54,55,59,
%U A029157 64,66,70,76,78,82,89,91,96,103,106,111,119,122,128,136
%N A029157 Expansion of 1/((1-x^2)(1-x^3)(1-x^7)(1-x^9)).
%Y A029157 Adjacent sequences: A029154 A029155 A029156 this_sequence A029158 A029159 A029160
%Y A029157 Sequence in context: A083403 A114091 A070867 this_sequence A031437 A029145 A097986
%K A029157 nonn
%O A029157 0,7
%A A029157 njas
%I A031437
%S A031437 1,1,1,2,2,2,4,3,4,9,14,33,839,2041,22192
%N A031437 Number of nonisomorphic regular linear spaces RLIN(n).
%D A031437 A. Betten and D. Betten: Regular Linear Spaces, Beitraege Algebra Geometrie 38 (1): 111-124, 1997.
%D A031437 A. Betten and D. Betten: The proper linear spaces on 17 points, Discrete Applied Mathematics, Volume 95, no. 1-3, 1999, pp. 83-108.
%D A031437 CRC Handbook of Combinatorial Designs, in the article by Gronau, Mullin and Pietsch.
%Y A031437 Cf. A001200, A031436.
%Y A031437 Adjacent sequences: A031434 A031435 A031436 this_sequence A031438 A031439 A031440
%Y A031437 Sequence in context: A114091 A070867 A029157 this_sequence A029145 A097986 A096445
%K A031437 nonn,nice
%O A031437 0,4
%A A031437 Anton Betten (Anton.Betten(AT)uni-bayreuth.de)
%I A029145
%S A029145 1,0,1,1,1,2,2,2,4,3,5,5,6,7,8,9,11,11,14,14,17,18,20,22,
%T A029145 25,26,30,31,35,37,41,43,48,50,55,58,63,66,72,75,82,85,
%U A029145 92,96,103,108,115,120,129,133,143,148,157,164,173,180
%N A029145 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^8)).
%Y A029145 Adjacent sequences: A029142 A029143 A029144 this_sequence A029146 A029147 A029148
%Y A029145 Sequence in context: A070867 A029157 A031437 this_sequence A097986 A096445 A125915
%K A029145 nonn
%O A029145 0,6
%A A029145 njas
%I A097986
%S A097986 1,1,2,2,2,4,3,5,5,7,6,12,9,13,15,20,18,28,26,37,39,47,49,71,68,85,94,
%T A097986 117,120,159,160,201,216,257,277,348,357,430,470,562,592,720,758,901,
%U A097986 981,1134,1220,1457,1542,1798,1952,2250,2419,2819,3023,3482,3773,4291
%N A097986 Number of partitions of n into distinct parts, each of which has a part which divides every part in the partition.
%F A097986 a(n) = Sum_{d|n} A025147(d-1). G.f.: Sum(x^k*Product(1+x^(k*i), i=2..infinity), k=1..infinity).
%t A097986 Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (from Robert G. Wilson v Nov 01 2004)
%Y A097986 Cf. A083710.
%Y A097986 Adjacent sequences: A097983 A097984 A097985 this_sequence A097987 A097988 A097989
%Y A097986 Sequence in context: A029157 A031437 A029145 this_sequence A096445 A125915 A071472
%K A097986 easy,nonn
%O A097986 1,3
%A A097986 Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 23 2004
%E A097986 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 01 2004
%I A096445
%S A096445 1,1,2,2,2,4,4
%N A096445 Number of reduced primitive positive definite binary quadratic forms of determinant n^2.
%C A096445 Equivalently, of discriminant -4n^2.
%H A096445 J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd edition, 1999, see Table 15.1.
%Y A096445 Equals A096446(n^2). Cf. A006374.
%Y A096445 Adjacent sequences: A096442 A096443 A096444 this_sequence A096446 A096447 A096448
%Y A096445 Sequence in context: A031437 A029145 A097986 this_sequence A125915 A071472 A064135
%K A096445 nonn,more
%O A096445 1,3
%A A096445 njas, Aug 11 2004
%I A125915
%S A125915 1,0,1,2,2,2,4,4,1,1
%N A125915 Sprague-Grundy values for octal game .147.
%C A125915 Octal games .544, .545, .546 and .547 also have values a(n).
%C A125915 The sequence is eventually periodic with period 8. The last exception is at n=2.
%D A125915 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4, p. 104.
%Y A125915 Adjacent sequences: A125912 A125913 A125914 this_sequence A125916 A125917 A125918
%Y A125915 Sequence in context: A029145 A097986 A096445 this_sequence A071472 A064135 A105080
%K A125915 nonn
%O A125915 1,4
%A A125915 Richard Sabey (richardsabey(AT)hotmail.co.uk), Jan 24 2007
%I A071472
%S A071472 1,1,0,2,2,2,4,4,1,1,1,3,2,2,4,4,4,6,6,6,2,1,1,1,5,7,6,6,8,8,1,1,1,
%T A071472 2,6,5,5,5,8,1
%N A071472 Sprague-Grundy values for octal game .156y.
%D A071472 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982; see Chapter 4.
%Y A071472 Adjacent sequences: A071469 A071470 A071471 this_sequence A071473 A071474 A071475
%Y A071472 Sequence in context: A097986 A096445 A125915 this_sequence A064135 A105080 A117728
%K A071472 nonn
%O A071472 1,4
%A A071472 njas and Sue Pope (pope(AT)research.att.com), May 29 2002
%I A064135
%S A064135 2,2,2,4,4,2,2,2,4,4,4,8,2,4,8,8,32,8,2,2,4,8,8,16,2,8,4,4,4,4,8,2,16,
%T A064135 4,8,4,8,8,4,16,4,4,4,8,4,8,8
%N A064135 Number of divisors of 10^n + 1 that are relatively prime to 10^m + 1 for all 0 < m < n.
%H A064135 Sam Wagstaff, Cunningham Project, Factorizations of 10^n-1, n<=330
%t A064135 a = {1}; Do[ d = Divisors[ 10^n + 1 ]; l = Length[ d ]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[ a, d[ [ k ] ] ] ] == {1}, c++ ]; k++ ]; Print[ c ]; a = Union[ Flatten[ Append[ a, Transpose[ FactorInteger[ 10^n + 1 ] ][ [ 1 ] ] ] ] ], {n, 0, 46} ]
%Y A064135 Adjacent sequences: A064132 A064133 A064134 this_sequence A064136 A064137 A064138
%Y A064135 Sequence in context: A096445 A125915 A071472 this_sequence A105080 A117728 A130453
%K A064135 nonn
%O A064135 0,1
%A A064135 Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 10 2001
%I A105080
%S A105080 2,2,2,4,4,2,3,4,4,5,4,5,4,4,4,4,5,3,5,4,5,3,4,8,4,3,6,5,5,3,7,5,3,5,1,
%T A105080 4,6,5
%N A105080 Number of distinct prime divisors of 10000^n - 3.
%H A105080 Makoto Kamada, Factorizations of 999...997.
%F A105080 A105080(n) = A105068(4n) = A105069(2n). - Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 08 2005
%e A105080 The number of distinct prime divisors of 10000^1 - 3 is 2, so the first term is 2.
%Y A105080 Cf. A089675, A105068, A105069.
%Y A105080 Adjacent sequences: A105077 A105078 A105079 this_sequence A105081 A105082 A105083
%Y A105080 Sequence in context: A125915 A071472 A064135 this_sequence A117728 A130453 A070705
%K A105080 nonn
%O A105080 1,1
%A A105080 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 06 2005
%E A105080 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 08 2005
%I A117728
%S A117728 1,2,2,2,4,4,2,4,5,4,6,4,4,8,4,4,8,6,6,8,8,4,6,8,5,12,8,4,12,8,6,8,8,
%T A117728 8,12,10,4,12,8,8,16,8,6,12,12,8,10,8,9,14,12,8,12,16,8,16,8,4,18,8,12,
%U A117728 16,10,8,16,16,6,16,16,8,14,12,8,20,14,12,16,8,10,16,17,8,18,16,8,20,12
%N A117728 A117726(n)/2.
%Y A117728 Adjacent sequences: A117725 A117726 A117727 this_sequence A117729 A117730 A117731
%Y A117728 Sequence in context: A071472 A064135 A105080 this_sequence A130453 A070705 A101909
%K A117728 nonn
%O A117728 1,2
%A A117728 njas, Apr 14 2006
%I A130453
%S A130453 1,2,2,2,4,4,2,4,8,8,2,4,8,16,16,2,4,8,16,32,32,2,4,8,16,32,64,64,2,4,8,
%T A130453 16,32,64,128,128
%N A130453 A097806 * A059268.
%C A130453 Row sums = A033484: (1, 4, 10, 22, 46, 94,...). A130459 = A059268 * A097806.
%F A130453 A097806 * A059268 as infinite lower triangular matrices.
%e A130453 First few rows of the triangle are:
%e A130453 1;
%e A130453 2, 2;
%e A130453 2, 4, 4;
%e A130453 2, 4, 8, 8;
%e A130453 2, 4, 8, 16, 16;
%e A130453 2, 4, 8, 16, 32, 32;
%e A130453 ...
%Y A130453 Cf. A097806, A059268, A033484, A130459.
%Y A130453 Adjacent sequences: A130450 A130451 A130452 this_sequence A130454 A130455 A130456
%Y A130453 Sequence in context: A064135 A105080 A117728 this_sequence A070705 A101909 A130872
%K A130453 nonn,tabl
%O A130453 1,2
%A A130453 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 26 2007
%I A070705
%S A070705 2,2,2,4,4,3,1,4,8,5,14,15,5,18,1,20,16,2,15,15,8,21,29,21,16,32,29,23,
%T A070705 22,30,54,71,37,7,37,43,45,30,36,77,100,72,64,7,56,33,42,54,132,18,90,
%U A070705 156,91,29,86,149,139,111,112,96,62,5,204,103,41,197,81,218,128,238,58
%N A070705 LCM of first n prime powers modulo next prime power.
%F A070705 a(n)=A051451(n)[mod A000961(n+1)]=A051126(A051451(n), A000961(n-1)).
%e A070705 The case n=7 implies a(7)=A051451(7)[mod A000961(8)]=LCM(2,3,4,5,7,8,9)[mod 11]=2520 [mod 11]=1.
%Y A070705 Cf. A051451, A000961.
%Y A070705 Adjacent sequences: A070702 A070703 A070704 this_sequence A070706 A070707 A070708
%Y A070705 Sequence in context: A105080 A117728 A130453 this_sequence A101909 A130872 A087627
%K A070705 nonn
%O A070705 1,1
%A A070705 Lekraj Beedassy (blekraj(AT)yahoo.com), May 15 2002
%E A070705 More terms from Don Reble (djr(AT)nk.ca), May 16 2002
%I A101909
%S A101909 1,2,2,2,4,4,3,5,4,4,6,6,6,7,7,7,8,9,9,10,10,9,10,9,10,12,12,13,14,13,
%T A101909 12,13,14,13,15,14,13,15,15,15,16,16,16,17,17,18,18,19,19,21,20,19,20,
%U A101909 19,18,19,19,20,21,22,23,23,24,23,24,24,24,26,25,25,27,27,27,28,27,26
%N A101909 Number of primes between 2n and 4n.
%H A101909 T. D. Noe, Table of n, a(n) for n=1..1000
%t A101909 f[n_] := PrimePi[4n] - PrimePi[2n]; Table[ f[n], {n, 76}] (from Robert G. Wilson v Feb 10 2005)
%o A101909 (PARI) bet2n4n(n) = { local(c,x,y); forstep(x=2,n,2, c=0; forprime(y=x+1,x+x-1, c++; ); print1(c",") ) }
%Y A101909 Cf. A101947, A101983, A101984, A101985.
%Y A101909 Adjacent sequences: A101906 A101907 A101908 this_sequence A101910 A101911 A101912
%Y A101909 Sequence in context: A117728 A130453 A070705 this_sequence A130872 A087627 A096491
%K A101909 easy,nonn
%O A101909 1,2
%A A101909 Cino Hilliard (hillcino368(AT)gmail.com), Jan 28 2005
%I A130872
%S A130872 1,1,1,2,2,2,4,4,3,5,5,5,7,7,6,9,10,8,11,11,10,15,14,12,16,17,15,19,19,
%T A130872 17,22,24,20,25,27,23,30,29,26,32,34,30,36,38,33,40,43,37,44,47,41,50,
%U A130872 52,45,53,55,50,58,62,54,63,70,59,68,71,64,75,79,70,79,85,77,85,89,81
%N A130872 Number of ways to write 8n - 1 as a sum of 7 (not necessarily distinct) odd perfect squares.
%C A130872 Since all odd perfect squares are congruent to 1 mod 8, it is not possible to express any number congruent to 7 mod 8 as a sum of fewer than 7 odd perfect squares.
%e A130872 a(2) = 1 because 15 = 9 + 1 + 1 + 1 + 1 + 1 + 1 is the only such representation. a(4) = 2 because 31 = 25 + 1 + 1 + 1 + 1 + 1 + 1 = 9 + 9 + 9 + 1 + 1 + 1 + 1.
%p A130872 A130872recur := proc(n,jmin,N) local a,s ; if N =1 then if n mod 2 = 1 and issqr(n) and n>=jmin^2 then RETURN(1) ; else RETURN(0) ; fi ; else a := 0 ; for s from 2*floor(jmin/2)+1 to floor(sqrt(n)) by 2 do a := a+A130872recur(n-s^2,s,N-1) ; od ; RETURN(a) ; fi ; end: A130872 := proc(n) option remember: A130872recur(8*n-1,1,7) ; end: seq(A130872(n),n=1..100); - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 02 2007
%Y A130872 Adjacent sequences: A130869 A130870 A130871 this_sequence A130873 A130874 A130875
%Y A130872 Sequence in context: A130453 A070705 A101909 this_sequence A087627 A096491 A106160
%K A130872 nonn
%O A130872 1,4
%A A130872 Joel Lewis (jblewis(AT)post.harvard.edu), Jul 24 2007
%E A130872 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 02 2007
%I A087627
%S A087627 1,2,2,2,4,4,3,6,6,4,8,8,5,10,10,6,12,12,7,14,14,8,16,16,9,18,18,10,20,
%T A087627 20,11,22,22,12,24,24,13,26,26,14,28,28,15,30,30,16,32,32,17,34,34,18,
%U A087627 36,36,19,38,38,20,40,40,21,42,42,22,44,44,23,46,46,24,48,48,25,50,50
%N A087627 Count ...n,2n,2n...
%C A087627 a(n)=#{0<=k<=n: 3 divides k(n-k) }
%F A087627 a(n)=sum{k=0..n, if (mod(k(n-k), 3)=0, 1, 0) }
%Y A087627 Adjacent sequences: A087624 A087625 A087626 this_sequence A087628 A087629 A087630
%Y A087627 Sequence in context: A070705 A101909 A130872 this_sequence A096491 A106160 A007614
%K A087627 easy,nonn
%O A087627 0,2
%A A087627 Paul Barry (pbarry(AT)wit.ie), Sep 14 2003
%I A096491
%S A096491 1,2,2,2,4,4,4,4,3,6,6,6,6,6,6,4,8,8,8,8,8,8,8,8,5,10,10,10,10,10,10,10,
%T A096491 10,10,10,6,12,12,12,12,12,12,12,12,12,12,12,12,7,14,14,14,14,14,14,14,
%U A096491 14,14,14,14,14,14,14,8,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16
%N A096491 Largest term in continued fraction period of square root of n.
%e A096491 n=127: the period={3,1,2,2,7,11,7,2,2,1,3,22}, max=a[127]=22;
%t A096491 u=1;Do[s=Max[Last[ContinuedFraction[n^(1/2)]]];tc[[u]]=s;u=u+1, {n, 1, m}]
%Y A096491 Cf. A003285, A013646.
%Y A096491 Adjacent sequences: A096488 A096489 A096490 this_sequence A096492 A096493 A096494
%Y A096491 Sequence in context: A101909 A130872 A087627 this_sequence A106160 A007614 A113402
%K A096491 cofr,nonn
%O A096491 1,2
%A A096491 Labos E. (labos(AT)ana.sote.hu), Jun 29 2004
%I A106160
%S A106160 2,2,2,4,4,4,4,4,6,6,6,8,6,8,8,8
%N A106160 Highest minimal Hamming distance of Hermitian Type IV self-dual codes over GF(2) X GF(2) and length 2n.
%D A106160 K. Betsumiya, T. A. Gulliver and M. Harada, Extremal self-dual codes over F_2 X \F_2, Designs, Codes Crypt. 28 (2003), 171-186.
%D A106160 K. Betsumiya and M. Harada, Optimal self-dual codes over F_2 X F_2 with respect to the Hamming weight, IEEE Trans. Inform. Theory 50 (2004), 356-358.
%D A106160 W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
%H A106160 G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
%Y A106160 Adjacent sequences: A106157 A106158 A106159 this_sequence A106161 A106162 A106163
%Y A106160 Sequence in context: A130872 A087627 A096491 this_sequence A007614 A113402 A054861
%K A106160 nonn
%O A106160 1,1
%A A106160 njas, May 08 2005
%I A007614
%S A007614 1,1,2,2,2,4,4,4,4,6,6,6,6,8,8,8,8,8,10,10,12,12,12,12,12,12,16,16,
%T A007614 16,16,16,16,18,18,18,18,20,20,20,20,20,22,22,24,24,24,24,24,24,24,
%U A007614 24,24,24,28,28,30,30,32,32,32,32,32,32,32,36,36,36,36,36,36,36,36
%N A007614 All values attained by the phi(n) function, in ascending order.
%C A007614 Write down phi(1), phi(2), phi(3), ..., then sort this list. Of course the list before sorting is simply sequence A000010.
%H A007614 T. D. Noe, Table of n, a(n) for n=1..10000
%Y A007614 Corresponding values of n are given by A032447. Cf. A000010.
%Y A007614 Adjacent sequences: A007611 A007612 A007613 this_sequence A007615 A007616 A007617
%Y A007614 Sequence in context: A087627 A096491 A106160 this_sequence A113402 A054861 A086227
%K A007614 nonn,easy,nice
%O A007614 1,3
%A A007614 wnissen(AT)tfn.net (Walter Nissen)
%I A113402
%S A113402 1,1,1,2,2,2,4,4,4,4,8,8,8,8,8,16,16,16,16,16,16,32,32,32,32,32,32,32,
%T A113402 64,64,64,64,64,64,64,64,128,128,128,128,128,128,128,128,128,256,256,
%U A113402 256,256,256,256,256,256,256,256,512,512,512,512,512,512,512,512,512
%N A113402 Algebraic degree of Cos[Pi/n] for constructible n-gons (A003401).
%H A113402 Eric Weisstein's World of Mathematics, Trigonometry Angles
%Y A113402 Cf. A003401, A055034, A113401.
%Y A113402 Adjacent sequences: A113399 A113400 A113401 this_sequence A113403 A113404 A113405
%Y A113402 Sequence in context: A096491 A106160 A007614 this_sequence A054861 A086227 A079438
%K A113402 nonn,easy,nice
%O A113402 1,4
%A A113402 Eric Weisstein (eric(AT)weisstein.com), Oct 28, 2005
%I A054861
%S A054861 0,0,0,1,1,1,2,2,2,4,4,4,5,5,5,6,6,6,8,8,8,9,9,9,10,10,10,13,13,13,14,
%T A054861 14,14,15,15,15,17,17,17,18,18,18,19,19,19,21,21,21,22,22,22,23,23,23,
%U A054861 26,26,26,27,27,27,28,28,28,30,30,30,31,31,31,32,32,32,34,34,34,35,35
%N A054861 Highest power of 3 dividing n!.
%C A054861 Also the number of trailing zeros in the base-3 representation of n!. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
%C A054861 Also the highest power of 6 dividing n!. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%H A054861 T. D. Noe, Table of n, a(n) for n=0..1000
%F A054861 a(n) = floor[n/3] + floor[n/9] + floor[n/27] + floor[n/81] + ....
%F A054861 G.f.: g(x)=sum{k>0, x^(3^k)/(1-x^(3^k))}/(1-x). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 18 2007
%F A054861 a(n)=sum{3<=k<=n, sum{j>=3,j|k, floor(log_3(j))-floor(log_3(j-1))}}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A054861 G.f.: g(x)=L[b(k)](x)/(1-x) where L[b(k)](x)=sum{k>=0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k)=1, if k>1 is a power of 3, else b(k)=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A054861 G.f.: g(x)=sum{k>0, c(k)*x^k}/(1-x), where c(k)=sum{j>1,j|k, floor(log_3(j))-floor(log_3(j-1))}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007
%F A054861 Recurrence: a(n)=floor(n/3)+a(floor(n/3)); a(3*n)=n+a(n); a(n*3^m)=n*(3^m-1)/2+a(n). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 a(k*3^m)=k*(3^m-1)/2, for 0<=k<3, m>=0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 Asymptotic behavior: a(n)=n/2+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 a(n)<=(n-1)/2; equality holds for powers of 3. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 a(n)>=(n-2)/2-floor(log_3(n)); equality holds for n=3^m-1, m>0. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 lim inf (n/2-a(n))=1/2, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 lim sup (n/2-log_3(n)-a(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%F A054861 lim sup (a(n+1)-a(n)-log_3(n))=0, for n-->oo. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 14 2007
%t A054861 Table[t = 0; p = 3; While[s = Floor[n/p]; t = t + s; s > 0, p *= 2]; t, {n, 0, 100} ]
%Y A054861 a(n+1)=sum(k=1,n, A007949(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 24 2002
%Y A054861 a(n)=(n-A053735(n))/2
%Y A054861 Cf. A011371 for analogue involving powers of 2. See also A027868.
%Y A054861 See A004128 for a(3n).
%Y A054861 Cf. A054895, A067080, A098844, A132027.
%Y A054861 Adjacent sequences: A054858 A054859 A054860 this_sequence A054862 A054863 A054864
%Y A054861 Sequence in context: A106160 A007614 A113402 this_sequence A086227 A079438 A123050
%K A054861 easy,nonn
%O A054861 0,7
%A A054861 Henry Bottomley (se16(AT)btinternet.com), May 22 2000
%I A086227
%S A086227 1,2,2,2,4,4,4,6,4,6,8,6,8,8,8,8,12,10,8,16,12,12,16,10,12,18,16,14,16,
%T A086227 16,16,24,16,16,24,18,20,24,16,20,32,22,24,24,24,24,32,28,20,32,24,26,36,
%U A086227 24,32,40,28,30,32,30,32,48,32,24,48,34,32,48,32,36,48,36,36,40,40,48,48
%V A086227 -1,2,-2,2,-4,4,-4,6,-4,6,-8,6,-8,8,-8,8,-12,10,-8,16,-12,12,-16,10,-12,18,-16,14,-16,
%W A086227 16,-16,24,-16,16,-24,18,-20,24,-16,20,-32,22,-24,24,-24,24,-32,28,-20,32,-24,26,-36,
%X A086227 24,-32,40,-28,30,-32,30,-32,48,-32,24,-48,34,-32,48,-32,36,-48,36,-36,40,-40,48,-48
%N A086227 a(n)=1/(4i)*sum( i^k*tan(k*Pi/4/n)) where 1<=k<=n and (k,n)=1.
%o A086227 (PARI) a(n)=round(real(1/4/I*sum(k=1,4*n,(I^k)*tan(Pi/4/n*if(gcd(k,n)-1,0,k)))))
%Y A086227 Adjacent sequences: A086224 A086225 A086226 this_sequence A086228 A086229 A086230
%Y A086227 Sequence in context: A007614 A113402 A054861 this_sequence A079438 A123050 A113694
%K A086227 sign
%O A086227 2,2
%A A086227 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 28 2003
%I A079438
%S A079438 1,1,2,2,2,4,4,4,6,6,6,8,8,8,12,12,12,14,16,16,18,18,22,24,24,24,28,28,
%T A079438 28,30,34,34,36,36,38,40,40,40,46,46,46,48,50,50,52,52,56,58,58,58,62,
%U A079438 62,62,64,68,68,70,70,72,74,74,74,80,80,80,82,84,84,86,86,90,92,92,92
%N A079438 Number of rooted general plane trees which are symmetric and will stay symmetric also after the underlying plane binary tree has been reflected, i.e. number of integers i in range [A014137(n-1)..A014138(n-1)] such that A057164(i)=i and A057164(A057163(i)) = A057163(i).
%C A079438 Also number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i)=A069787(i)=i, i.e. the size of the intersect of fixed points of permutations A057164 and A069787 in the same range.
%D A079438 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%H A079438 A. Karttunen, C-program for counting the initial terms of this sequence (empirically)
%H A079438 A. Karttunen, Illustration of initial terms for trees of sizes n=2..18
%F A079438 a(0)=a(1)=1, a(n) = 2*(floor((n+1)/3) + (if n>=14) (floor((n-10)/4)+floor((n-14)/8))) [This is the correct formula if the conjecture given in A080070 is true, otherwise it is only a lower bound, although known to be exact for up to very high values of n.]
%p A079438 A079438 := n -> `if`((n<2),1,2*(floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0)));
%Y A079438 From n>= 2 onward A079440(n) = a(n)/2.
%Y A079438 Occurs in A073202 as row 13373289. Cf. A079437, A079439, A079442, A080070.
%Y A079438 Adjacent sequences: A079435 A079436 A079437 this_sequence A079439 A079440 A079441
%Y A079438 Sequence in context: A113402 A054861 A086227 this_sequence A123050 A113694 A086159
%K A079438 nonn
%O A079438 0,3
%A A079438 Antti Karttunen (Firstname.Surname(AT)iki.fi) Jan 27 2003
%I A123050
%S A123050 1,1,2,2,2,4,4,4,6,6,6,8,8,8,12,12,12,14,16,16,18,18,22,24,24,24,30,30,
%T A123050 30,32,38,38,40,40,46,48,48,48,58,58,58,60,68,68,70,70,80,82,82,82,94,
%U A123050 94,94,96,108,108,110,110,122,124,124,124,140,140,140,142,156,156,158
%N A123050 Conjectured number of ordered trees on n edges for which the conjugate and transpose commute.
%C A123050 The conjugate of an ordered tree is given by flipping it over, while its transpose is given by flipping over the corresponding binary tree. A list of ordered trees for which the conjugate and transpose commute, counted by this sequence, is given in Exercise 17, Sec. 7.2.1.6 of the Knuth reference. (Knuth deletes the root from an ordered tree and works with the resulting forest instead.) This list is complete provided a certain set of ordered trees contains no self-conjugate members other than the "obvious" ones.
%C A123050 The set in question consists of all trees generated by repeatedly applying the following two productions to the one-edge tree: (i) T -> plant(T) (i.e. add an edge to the root to obtain a new root), and (ii) T -> add left root edge to the transpose of the conjugate of T. Computational evidence suggests that this proviso does indeed hold.
%D A123050 D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006).
%H A123050 D. E. Knuth,Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.
%F A123050 a(0)=a(1)=1, a(n) = 2(Floor[(n+1)/3] + Sum[Max[0,Floor[(n-(8k+2))/4]],{k,1,(n-2)/8}]) for n>=2. GF: 1 + x + 2x^2/((1-x)(1-x^3)) + 2x^14/((1-x)*(1-x^4)*(1-x^8))
%t A123050 a[0]=a[1]=1; a[n_]/;n>=2 := 2(Floor[(n+1)/3] + Sum[Max[0,Floor[(n-(8k+2))/4]],{k,1,(n-2)/8}]); Table[a[n],{n,0,90}]
%Y A123050 This sequence updates the lower bound conjectured in A079438.
%Y A123050 Adjacent sequences: A123047 A123048 A123049 this_sequence A123051 A123052 A123053
%Y A123050 Sequence in context: A054861 A086227 A079438 this_sequence A113694 A086159 A029048
%K A123050 nonn
%O A123050 0,3
%A A123050 David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
%I A113694
%S A113694 0,0,0,2,2,2,4,4,4,6,6,6,8,8,9,1,1,1,3,3,3,5,5,5,7,7,8
%N A113694 Decimal expansion of 10/44955.
%C A113694 We can get this sequence from 10/44955 or from Sqrt[128128128128128128128128128128128128128128128128128128] =24Sqrt[222444666889111333555778000222444666889111333555778], where Sqrt is the square root.
%t A113694 n = 17; Sqrt[128*Apply[Plus, Table[(10^3)^k, {k, 0, n}]]]
%Y A113694 Cf. A021895, A021085.
%Y A113694 Adjacent sequences: A113691 A113692 A113693 this_sequence A113695 A113696 A113697
%Y A113694 Sequence in context: A086227 A079438 A123050 this_sequence A086159 A029048 A086160
%K A113694 easy,cons,nonn
%O A113694 0,4
%A A113694 Daisuke Minematsu and Ryohei Miyadera (miyadera1272000(AT)yahoo.co.jp), Jan 17 2006
%I A086159
%S A086159 1,1,1,2,2,2,4,4,4,6,6,6,9,9,9,12,12,12,16,16,16,20,20,20,25,25,25,30,
%T A086159 30,30,36,36,36,42,42,42,49,49,49,56,56,56,64,64,64,72,72,72,81,81,81,
%U A086159 90,90,90,100,100,100,110
%N A086159 Number of partitions of n into the first three triangular numbers, 1, 3, and 6.
%H A086159 Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
%F A086159 G.f.: 1/((1-x)*(1-x^3)*(1-x^6))
%Y A086159 Cf. A008620.
%Y A086159 Adjacent sequences: A086156 A086157 A086158 this_sequence A086160 A086161 A086162
%Y A086159 Sequence in context: A079438 A123050 A113694 this_sequence A029048 A086160 A029047
%K A086159 nonn
%O A086159 0,4
%A A086159 Jan Snellman (Jan.Snellman(AT)math.su.se), Aug 25 2003
%I A029048
%S A029048 1,1,1,2,2,2,4,4,4,6,6,7,10,10,11,14,14,16,20,20,22,26,
%T A029048 27,30,35,36,39,44,46,50,56,58,62,69,72,77,85,88,93,102,
%U A029048 106,112,122,126,133,144,149,157,169,174,183,196,202,212
%N A029048 Expansion of 1/((1-x)(1-x^3)(1-x^6)(1-x^11)).
%Y A029048 Adjacent sequences: A029045 A029046 A029047 this_sequence A029049 A029050 A029051
%Y A029048 Sequence in context: A123050 A113694 A086159 this_sequence A086160 A029047 A007294
%K A029048 nonn
%O A029048 0,4
%A A029048 njas
%I A086160
%S A086160 1,1,1,2,2,2,4,4,4,6,7,7,10,11,11,14,16,16,20,22,23,27,30,31,36,39,41
%N A086160 Duplicate of A029047.
%H A086160 Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
%Y A086160 Adjacent sequences: A086157 A086158 A086159 this_sequence A086161 A086162 A086163
%Y A086160 Sequence in context: A113694 A086159 A029048 this_sequence A029047 A007294 A053282
%K A086160 dead
%O A086160 0,4
%I A029047
%S A029047 1,1,1,2,2,2,4,4,4,6,7,7,10,11,11,14,16,16,20,22,23,27,
%T A029047 30,31,36,39,41,46,50,52,59,63,66,73,78,81,90,95,99,108,
%U A029047 115,119,130,137,142,153,162,167,180,189,196,209,220,227
%N A029047 Expansion of 1/((1-x)(1-x^3)(1-x^6)(1-x^10)).
%C A029047 Number of partitions of n into the first four triangular numbers, 1, 3, 6, and 10.
%Y A029047 Cf. A008620.
%Y A029047 Adjacent sequences: A029044 A029045 A029046 this_sequence A029048 A029049 A029050
%Y A029047 Sequence in context: A086159 A029048 A086160 this_sequence A007294 A053282 A001584
%K A029047 nonn
%O A029047 0,4
%A A029047 njas
%I A007294 M0234
%S A007294 1,1,1,2,2,2,4,4,4,6,7,7,10,11,11,15,17,17,22,24,25,32,35,36,44,
%T A007294 48,50,60,66,68,81,89,92,107,117,121,141,153,159,181,197,205,233,
%U A007294 252,262,295,320,332,372,401,417,465,501,520,575,619,645,710,763
%N A007294 Number of partitions of n into nonzero triangular numbers.
%C A007294 Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.
%C A007294 a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.
%C A007294 Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2008
%D A007294 G. E. Andrews, MacMahon's partition analysis: II, Fundamental theorems, Annals of Combinatorics, 4 (2000), 327-338.
%H A007294 T. D. Noe, Table of n, a(n) for n = 0..1000
%H A007294 James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
%H A007294 Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
%F A007294 Product('(1-z^binomial(k, 2))^(-1)', 'k'=2..infinity);
%F A007294 G.f.: Product ( 1 - x^(k(k-1)/2))^(-1). - Les Reid, Jul 26, 2002
%F A007294 For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Aug 26 2003
%e A007294 6=3+3=3+1+1+1=1+1+1+1+1+1 so a(6) = 4.
%e A007294 a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.
%t A007294 CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
%Y A007294 Cf. A000217, A051533, A000294.
%Y A007294 Cf. A102462.
%Y A007294 Adjacent sequences: A007291 A007292 A007293 this_sequence A007295 A007296 A007297
%Y A007294 Sequence in context: A029048 A086160 A029047 this_sequence A053282 A001584 A112801
%K A007294 nonn
%O A007294 0,4
%A A007294 njas, Mira Bernstein (mira(AT)math.berkeley.edu)
%E A007294 Additional comments from Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Jun 17 2001
%I A053282
%S A053282 0,1,1,2,2,2,4,4,4,6,7,8,10,11,12,16,18,20,24,26,30,36,40,44,52,58,64,
%T A053282 74,82,91,104,116,128,144,159,176,198,218,240,268,294,324,360,394,432,
%U A053282 478,524,572,630,688,752,826,900,980,1072,1168,1270,1386,1505,1634
%N A053282 Coefficients of the '10th order' mock theta function psi(q)
%D A053282 Youn-Seo Choi, Tenth order mock theta functions in Ramanujan's lost notebook, Inventiones Mathematicae, 136 (1999) 497-569
%D A053282 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 9
%F A053282 G.f.: psi(q) = sum for n >= 0 of q^((n+1)(n+2)/2)/((1-q)(1-q^3)...(1-q^(2n+1)))
%t A053282 Series[Sum[q^((n+1)(n+2)/2)/Product[1-q^(2k+1), {k, 0, n}], {n, 0, 12}], {q, 0, 100}]
%Y A053282 Other '10th order' mock theta functions are at A053281, A053283, A053284.
%Y A053282 Adjacent sequences: A053279 A053280 A053281 this_sequence A053283 A053284 A053285
%Y A053282 Sequence in context: A086160 A029047 A007294 this_sequence A001584 A112801 A089873
%K A053282 nonn,easy
%O A053282 0,4
%A A053282 Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 19 1999
%I A001584 M0235 N0080
%S A001584 1,1,1,1,1,1,1,1,2,2,2,4,4,4,7,7,8,12,12,16,21,21,31,37,38,58,65,71,106,114,
%T A001584 135,191,201,257,341,359,485,605,652,904,1070,1202,1664,1894,2237,3029,3370,
%U A001584 4176,5464,6048,7779,9793,10963,14411,17492,20054,26507,31239,36924,48396
%N A001584 A generalized Fibonacci sequence.
%D A001584 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%D A001584 V. C. harris, Generalized Fibonacci sequences associated with a generalized Pascal triangle, Fib. Quart., 4 (1966), 241-248.
%H A001584 T. D. Noe, Table of n, a(n) for n=0..1000
%H A001584 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001584 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A001584 G.f.: (1+x+x^2-x^3-x^4-x^5)/(1-2*x^3+x^6-x^8).
%p A001584 A001584:=(z-1)*(z**2+z+1)**2/(z**4-z**3+1)/(z**4+z**3-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
%Y A001584 Cf. A017817.
%Y A001584 Adjacent sequences: A001581 A001582 A001583 this_sequence A001585 A001586 A001587
%Y A001584 Sequence in context: A029047 A007294 A053282 this_sequence A112801 A089873 A096323
%K A001584 nonn,easy
%O A001584 0,9
%A A001584 njas
%E A001584 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
%I A112801
%S A112801 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,2,2,2,4,4,4,8,7,8,11,11,13,15,16,18,23,
%T A112801 23,26,30,31,33,40,40,45,51,53,56,62,66,66,76,79,82,88,94,96,105,111,
%U A112801 111,124,127,132,141,145,148,164,166,170,180,187,187,206,204,208
%N A112801 Number of ways of representing 2n-1 as sum of three integers with 2 distinct prime factors.
%C A112801 Meng proves a remarkable generalization of the Goldbach-Vinogradov classical result that every sufficiently large odd integer N can be partitioned as the sum of three primes N = p1 + p2 + p3. The new proof is that every sufficiently large odd integer N can be partitioned as the sum of three integers N = a + b + c where each of a, b, c has k distinct prime factors for the same k.
%D A112801 Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
%F A112801 Number of ways of representing 2n-1 as sum of three semiprimes (A001358) or products of two powers of primes (A000961 except 1). Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 2. Number of ways of representing 2n-1 as a + b + c where A001221(a) = A001221(b) A001221(c) = 2.
%e A112801 a(14) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*14)-1 = 27 is 27 = 6 + 6 + 15 = (2*3) + (2*3) + (3*5).
%e A112801 a(16) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*16)-1 = 31 is 31 = 6 + 10 + 15 = (2*3) + (2*5) + (3*5).
%e A112801 a(17) = 2 because the two partitions into three integers each with 2 distinct prime factors of (2*17)-1 = 33 are 33 = 6 + 6 + 21 = 6 + 12 + 15.
%Y A112801 Cf. A000961, A001358, A112799, A112800, A112802.
%Y A112801 Adjacent sequences: A112798 A112799 A112800 this_sequence A112802 A112803 A112804
%Y A112801 Sequence in context: A007294 A053282 A001584 this_sequence A089873 A096323 A035682
%K A112801 nonn
%O A112801 1,17
%A A112801 Jonathan Vos Post (jvospost2(AT)yahoo.com) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 19 2005
%I A089873
%S A089873 1,1,2,2,2,4,4,4,10,6,10,12,16,20,36,28,56,46,104,84,166
%N A089873 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A071665/A071666.
%C A089873 The number of n-node binary trees fixed by the corresponding automorphism(s).
%H A089873 A. Karttunen, C-program for computing the initial terms of this sequence
%Y A089873 Adjacent sequences: A089870 A089871 A089872 this_sequence A089874 A089875 A089876
%Y A089873 Sequence in context: A053282 A001584 A112801 this_sequence A096323 A035682 A054543
%K A089873 nonn
%O A089873 0,3
%A A089873 Antti Karttunen (His_Firstname.His_Surname(AT)iki.fi), Nov 29 2003
%I A096323
%S A096323 1,2,2,2,4,4,5
%N A096323 a(n) is the maximum value such that an n-bit Gray (code) cycle can be found in which all runs have length >= a(n).
%C A096323 Exercise 62 in Knuth asks if a(8)=6. a(10) >= 8 (exercise 63) and is "almost certainly" 8 (Knuth).
%D A096323 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.1.
%Y A096323 Adjacent sequences: A096320 A096321 A096322 this_sequence A096324 A096325 A096326
%Y A096323 Sequence in context: A001584 A112801 A089873 this_sequence A035682 A054543 A029046
%K A096323 hard,nonn
%O A096323 1,2
%A A096323 Jud McCranie (j.mccranie(AT)comcast.net), Jun 28 2004
%I A035682
%S A035682 0,0,0,0,0,1,1,1,1,1,2,2,2,4,4,5,5,5,7,7,8,11,12,14,14,15,19,20,22,26,
%T A035682 29,34,35,37,43,46,51,57,63,72,76,81,91,98,107,117,128,144,153,163,
%U A035682 179,192,210,226,245,272,290,310,336,360,391,418,450,494,527,564,605
%N A035682 Number of partitions of n into parts 8k+1 and 8k+5 with at least one part of each type.
%Y A035682 Adjacent sequences: A035679 A035680 A035681 this_sequence A035683 A035684 A035685
%Y A035682 Sequence in context: A112801 A089873 A096323 this_sequence A054543 A029046 A035372
%K A035682 nonn
%O A035682 1,11
%A A035682 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A054543
%S A054543 2,2,2,4,4,5,5,12,13,41,110,172,248,309,3146,5919,21959,22299,30892,
%T A054543 401838,1719239,30576561,262313756,630913752,3242181301,3250783944,
%U A054543 13827502849,40152067840,137791590233,2514510232695,3217773878849
%N A054543 Engel series expansion (or "Egyptian product") for Catalan's constant G.
%D A054543 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 53-59.
%H A054543 S. R. Finch, Catalan's Constant
%H A054543 Index entries for sequences related to Engel expansions
%H A054543 Eric Weisstein's World of Mathematics, Engel Expansion
%H A054543 Eric Weisstein's World of Mathematics, Catalan's Constant
%Y A054543 Cf. A006784, A028254, A028257.
%Y A054543 Adjacent sequences: A054540 A054541 A054542 this_sequence A054544 A054545 A054546
%Y A054543 Sequence in context: A089873 A096323 A035682 this_sequence A029046 A035372 A035576
%K A054543 nonn
%O A054543 1,1
%A A054543 Jeppe Stig Nielsen (sequence(AT)jeppesn.dk), Apr 09 2000
%I A029046
%S A029046 1,1,1,2,2,2,4,4,5,7,7,8,11,11,13,16,17,19,23,24,27,31,
%T A029046 33,36,42,44,48,54,57,61,69,72,78,86,90,96,106,110,118,
%U A029046 128,134,142,154,160,170,182,190,200,215,223,235,250,260
%N A029046 Expansion of 1/((1-x)(1-x^3)(1-x^6)(1-x^8)).
%Y A029046 Adjacent sequences: A029043 A029044 A029045 this_sequence A029047 A029048 A029049
%Y A029046 Sequence in context: A096323 A035682 A054543 this_sequence A035372 A035576 A005859
%K A029046 nonn
%O A029046 0,4
%A A029046 njas
%I A035372
%S A035372 1,1,2,2,2,4,4,5,7,7,9,12,13,16,19,22,26,31,36,41,48,56,63,75,85,96,
%T A035372 112,126,143,165,184,210,238,267,302,340,381,428,480,538,599,672,748,
%U A035372 832,930,1031,1144,1275,1408,1562,1730,1910,2111,2332,2571,2834,3121
%N A035372 Number of partitions of n into parts 5k+1 or 5k+3.
%Y A035372 Adjacent sequences: A035369 A035370 A035371 this_sequence A035373 A035374 A035375
%Y A035372 Sequence in context: A035682 A054543 A029046 this_sequence A035576 A005859 A134318
%K A035372 nonn
%O A035372 1,3
%A A035372 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A035576
%S A035576 1,1,1,2,2,2,4,4,5,7,7,10,13,14,18,21,27,33,38,47,54,67,84,94,116,136,
%T A035576 160,200,229,270,322,372,455,530,615,728,840,1003,1174,1352,1589,1826,
%U A035576 2150,2514,2880,3359,3857,4475,5224,5965,6896,7914,9098,10551,12042
%N A035576 Number of partitions of n with equal number of parts congruent to each of 0, 2 and 4 (mod 5)
%Y A035576 Adjacent sequences: A035573 A035574 A035575 this_sequence A035577 A035578 A035579
%Y A035576 Sequence in context: A054543 A029046 A035372 this_sequence A005859 A134318 A104295
%K A035576 nonn
%O A035576 0,4
%A A035576 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%E A035576 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
%I A005859 M0236
%S A005859 1,1,1,1,1,1,2,2,2,4,4,5,7,8,10,16,25
%N A005859 The coding-theoretic function A(n,12,9).
%D A005859 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005859 E. M. Rains and N. J. A. Sloane, A(n,d,w) tables
%H A005859 Index entries for sequences related to A(n,d,w)
%Y A005859 Adjacent sequences: A005856 A005857 A005858 this_sequence A005860 A005861 A005862
%Y A005859 Sequence in context: A029046 A035372 A035576 this_sequence A134318 A104295 A008331
%K A005859 nonn,hard
%O A005859 9,7
%A A005859 njas
%I A134318
%S A134318 1,1,1,2,2,2,4,4,6,4,8,8,14,16,8,16,16,30,44,40,16,32,32,62,104,128,96,
%T A134318 32,64,64,126,228,336,352,224,64,128,128,254,480,792,1024,928,512,128,
%U A134318 256,256,510,988,1752,2608,2976,2368,1152,256
%N A134318 A007318 * A134317.
%C A134318 Row sums = A025192: (1, 2, 6, 18, 54, 162,...).
%F A134318 Binomial transform of A134317, as infinite lower triangular matrices.
%e A134318 First few rows of the triangle are:
%e A134318 1;
%e A134318 1, 1;
%e A134318 2, 2, 2;
%e A134318 4, 4, 6, 4;
%e A134318 8, 8, 14, 16, 8;
%e A134318 16, 16, 30, 44, 40, 16;
%e A134318 32, 32, 62, 104, 128, 96, 32;
%e A134318 ...
%Y A134318 Cf. A134317, A025192.
%Y A134318 Adjacent sequences: A134315 A134316 A134317 this_sequence A134319 A134320 A134321
%Y A134318 Sequence in context: A035372 A035576 A005859 this_sequence A104295 A008331 A097196
%K A134318 nonn,tabl
%O A134318 0,4
%A A134318 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 19 2007
%I A104295
%S A104295 1,2,2,2,4,4,6,6,4,6,6,2,6,6,2,4,14,2,2,6,6,2,4,4,4,6,10,6,6,4,4,10,2,6,
%T A104295 4,4,2,2,6,2,4,4,6,2,18,6,6,2,12
%N A104295 A104294(n)-A104293(n).
%F A104295 p[(p[n]+1)/2]- p[(p[n]-1)/2]
%Y A104295 Cf. A104293, A104294.
%Y A104295 Adjacent sequences: A104292 A104293 A104294 this_sequence A104296 A104297 A104298
%Y A104295 Sequence in context: A035576 A005859 A134318 this_sequence A008331 A097196 A132325
%K A104295 easy,nonn
%O A104295 2,2
%A A104295 Zak Seidov (zakseidov(AT)yahoo.com), Feb 28 2005
%I A008331
%S A008331 2,2,2,4,4,6,6,8,8,8,16,18,12,20,16,18,16,30,32,24,36,32,24,24,42,32,48,36,
%T A008331 40,36,64,40,44,48,40,72,78,80,48,56,48,72,64,96,60,80,104,96,72,88,72,
%U A008331 64,110,72,84,80,72,128,138,92,140,84,120,96,156,104,164,156,112,120,116
%N A008331 phi(p+1), p prime.
%p A008331 for i from 1 to 500 do if isprime(i) then print(phi(i+1)); fi; od;
%Y A008331 Cf. A000010.
%Y A008331 Adjacent sequences: A008328 A008329 A008330 this_sequence A008332 A008333 A008334
%Y A008331 Sequence in context: A005859 A134318 A104295 this_sequence A097196 A132325 A010238
%K A008331 nonn
%O A008331 2,1
%A A008331 njas
%I A097196
%S A097196 1,0,1,2,2,2,4,4,6,8,9,12,16,18,22,28,33,40,50,58,70,84,98,116,138,160,
%T A097196 188,222,256,298,348,400,463,536,614,706,812,926,1060,1212,1378,1568,1785,
%U A097196 2022,2292,2598,2932,3312,3740,4208,4736,5328,5978,6708,7522,8416,9416
%N A097196 G.f.: Product_{n >= 1} (1+q^(3*n))^4*(1-q^(3*n))^2/(1-q^(2*n)).
%C A097196 The right-hand side of an identity of G. N. Watson.
%D A097196 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 50, Eq. (25.4).
%Y A097196 Adjacent sequences: A097193 A097194 A097195 this_sequence A097197 A097198 A097199
%Y A097196 Sequence in context: A134318 A104295 A008331 this_sequence A132325 A010238 A089819
%K A097196 nonn
%O A097196 0,4
%A A097196 njas, Sep 17 2004
%I A132325
%S A132325 2,2,2,4,4,6,9,1,3,8,2,7,4,1,0,1,2,6,4,2,5,2,1,5,6,1,3,4,1,8,8,8,1,1,6,
%T A132325 0,7,4,9,5,0,1,4,9,3,5,1,5,5,1,8,5,6,7,1,5,7,5,9,1,6,4,7,4,0,6,6,5,0,6,
%U A132325 9,3,8,9,7,6,2,8,2,2,0,8,7,5,2,9,4,4,4,4,5,2,8,4,2,7,0,4,7,1,1,2,9,4,8
%N A132325 Decimal expansion of product{k>=0, 1+1/10^k)}.
%C A132325 Twice the constant A132326.
%F A132325 lim sup product{0<=k<=floor(log_10(n)), (1+1/floor(n/10^k))} for n-->oo.
%F A132325 lim sup A132271(n)/n^((1+log_10(n))/2) for n-->oo.
%F A132325 lim sup A132272(n)/n^((log_10(n)-1)/2) for n-->oo.
%F A132325 2*exp(sum{n>0, 10^(-n)*sum{k|n, -(-1)^k/k}})=2*exp(sum{n>0, A000593(n)/(n*10^n)}).
%F A132325 lim sup A132271(n+1)/A132271(n)=2.22446913827410126425215613418881160749501... for n-->oo.
%e A132325 2.22446913827410126425215613418881160749501...
%Y A132325 Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.
%Y A132325 Adjacent sequences: A132322 A132323 A132324 this_sequence A132326 A132327 A132328
%Y A132325 Sequence in context: A104295 A008331 A097196 this_sequence A010238 A089819 A059888
%K A132325 nonn,cons
%O A132325 1,1
%A A132325 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 20 2007
%I A010238
%S A010238 1,1,2,2,2,4,4,7,12,18
%N A010238 Maximal size of binary code of length n and asymmetric distance 3.
%D A010238 T. Etzion, New lower bounds for asymmetric and unidirectional codes, IEEE Trans. Inform. Theory, 37 (1991), 1696-1705.
%D A010238 J. H. Weber, Bounds and Constructions for Binary Block Codes Correcting Asymmetric or Unidirectional Errors, Ph. D. Thesis, Tech. Univ. Delft, 1989.
%D A010238 J. H. Weber, C. de Vroedt and D. E. Boekee, Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6, IEEE Trans. Inform. Theory, 34 (1988), 1321-1332.
%Y A010238 Adjacent sequences: A010235 A010236 A010237 this_sequence A010239 A010240 A010241
%Y A010238 Sequence in context: A008331 A097196 A132325 this_sequence A089819 A059888 A126602
%K A010238 nonn
%O A010238 1,3
%A A010238 njas
%I A089819
%S A089819 2,2,2,4,4,8,8,16,32,64,64,128,128,256,512,1024,1024,2048,2048,4096,
%T A089819 8192,16384,16384,32768,65536,131072,262144,524288,524288,1048576,
%U A089819 1048576,2097152,4194304,8388608,16777216,33554432,33554432,67108864
%N A089819 Number of subsets of {1,.., n} containing no primes.
%C A089819 a(n) = Product(2-A010051(k): 1<=k<=n) = A089818(n,0) = A000079(n) - A089820(n).
%F A089819 a(n) = 2^(n-pi(n)), with pi = A000720.
%e A089819 a(6)=8 subsets of {1,2,3,4,5,6} contain no prime: {1,4,6}, {4,6},
%e A089819 {1,6}, {1,4}, {6}, {4}, {1}, and the empty set.
%Y A089819 Cf. A089821, A089822.
%Y A089819 Adjacent sequences: A089816 A089817 A089818 this_sequence A089820 A089821 A089822
%Y A089819 Sequence in context: A097196 A132325 A010238 this_sequence A059888 A126602 A024681
%K A089819 nonn
%O A089819 1,1
%A A089819 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Nov 12 2003
%I A059888
%S A059888 2,2,2,4,4,10,2,8,12,40,6,108,6,42,40,48,30,100,6,332,10,22,30,376,26,
%T A059888 118,48,332,2,1436,6,448,54,222,88,7952,62,54,54,2680,6,698,30,476,
%U A059888 1476,222,14,7632,28,438
%N A059888 a(n)=|{m : multiplicative order of 6 mod m=n}|.
%C A059888 The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
%F A059888 a(n)=Sum_{ d divides n } mu(n/d)*tau(6^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
%Y A059888 Cf. A000005, A008683, A053449, A059499, A059885, A059886, A059888-A059892.
%Y A059888 Adjacent sequences: A059885 A059886 A059887 this_sequence A059889 A059890 A059891
%Y A059888 Sequence in context: A132325 A010238 A089819 this_sequence A126602 A024681 A007495
%K A059888 easy,nonn
%O A059888 1,1
%A A059888 Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 06 2001
%I A126602
%S A126602 2,2,2,4,4,12,20,16,24,64,96,144,128,320,384,512,1008,1296,1024,2700,
%T A126602 2592,4800
%N A126602 a(n) = the maximum possible number of positive divisors of S, if S = product of b(k)'s + product of c(k)'s, where the distinct positive integers <= n are partitioned into the two sets {b(k)} and {c(k)}.
%C A126602 a(1) = 2 because the product over the empty set is defined here as 1. So we have a(1) = number of divisors of (1+1). Terms calculated by Peter Pein and J K Haugland.
%e A126602 For n = 6 the maximum number of divisors occurs when S = 1*3*4*5 + 2*6 = 72. (This 12-divisor solution is not unique.) So a(6) is the number of positive divisors of 72, which is 12.
%p A126602 A126602 := proc(n) local bc,a,b,c,i,j,bL,S,bsiz ; a := 0 ; bc := {seq(i,i=1..n)} ; for bsiz from 0 to floor(n/2) do bL := combinat[choose](bc,bsiz) ; for i from 1 to nops(bL) do b := convert(op(i,bL),set) ; c := bc minus b ; if nops(b) = 0 then b := 1; else b := mul(j,j=b) ; fi ; if nops(c) = 0 then c := 1; else c := mul(j,j=c) ; fi ; S := numtheory[tau](c+b) ; a := max(a,S) ; od: od: RETURN(a) ; end: for n from 1 do A126602(n) ; od; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 2007
%Y A126602 Adjacent sequences: A126599 A126600 A126601 this_sequence A126603 A126604 A126605
%Y A126602 Sequence in context: A010238 A089819 A059888 this_sequence A024681 A007495 A122385
%K A126602 more,nonn
%O A126602 1,1
%A A126602 Leroy Quet (qq-quet(AT)mindspring.com), Jan 06 2007
%E A126602 2 more terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 2007
%I A024681
%S A024681 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,4,5,4,6,10,9,11,10,16,13,23,21,24,
%T A024681 32,34,40,44,47,48,50,60,69,72,80,78,88,104,118,117,131,127,136,149,157,164,
%U A024681 177,192,201,212,213,223,236,277,279,294,290,342,358,367,398,385,409,441
%N A024681 a(n) = number of ways p(n) is a sum of 3 odd nonprimes r,s,t satisfying 9 <= r < s < t.
%Y A024681 Adjacent sequences: A024678 A024679 A024680 this_sequence A024682 A024683 A024684
%Y A024681 Sequence in context: A089819 A059888 A126602 this_sequence A007495 A122385 A035002
%K A024681 nonn
%O A024681 1,18
%A A024681 Clark Kimberling (ck6(AT)evansville.edu)
%I A007495 M0237
%S A007495 1,1,2,2,2,4,5,4,8,8,7,11,8,13,4,11,12,8,12,2,13,7,22,2,8,13,26,4,26,29,
%T A007495 17,27,26,7,33,20,16,22,29,4,13,22,25,14,22,37,18,46,42,46,9,41,12,7,26,
%U A007495 42,24,5,44,53,52,58,29,22
%N A007495 Josephus problem: survivors.
%D A007495 Friend H. Kierstead, Jr., Computer Challenge Corner, J. Rec. Math., 10 (1977), see p. 124.
%H A007495 T. D. Noe, Table of n, a(n) for n=1..1000
%Y A007495 Cf. A032434
%Y A007495 Adjacent sequences: A007492 A007493 A007494 this_sequence A007496 A007497 A007498
%Y A007495 Sequence in context: A059888 A126602 A024681 this_sequence A122385 A035002 A032578
%K A007495 easy,nonn
%O A007495 1,3
%A A007495 njas, Robert G. Wilson v (rgwv(AT)rgwv.com)
%I A122385
%S A122385 1,2,2,2,4,5,5,3,3,8,9,6,10,11,4,4,14,6,16,10,8,18,19,5,5,22,9,12,11,8,
%T A122385 26,6,6,29,6,6,14,33,26,20,36,16,37,22,12,40,41,7,7,10,45,26,47,18,21,
%U A122385 15,38,22,52,27,54,55,8,8,58,59,60,34,39,12,19,12,65,66,15,29,69,70,71
%N A122385 Smallest m such that (n^2 mod m) = (m^2 mod n).
%C A122385 A122388(n) = n^2 mod a(n) = a(n)^2 mod n;
%C A122385 A122386(n) = a(a(n));
%C A122385 a(A122387(n)) = n, a(m) <> n for m < A122387(n).
%e A122385 10^2 mod 1 = 0 <> 1^2 mod 10 = 1,
%e A122385 10^2 mod 2 = 0 <> 2^2 mod 10 = 4,
%e A122385 10^2 mod 3 = 1 <> 3^2 mod 10 = 9,
%e A122385 10^2 mod 4 = 0 <> 4^2 mod 10 = 6,
%e A122385 10^2 mod 5 = 0 <> 5^2 mod 10 = 5,
%e A122385 10^2 mod 6 = 4 <> 6^2 mod 10 = 6,
%e A122385 10^2 mod 7 = 2 <> 7^2 mod 10 = 9,
%e A122385 10^2 mod 8 = 4 = 8^2 mod 10, therefore a(10) = 8.
%Y A122385 Adjacent sequences: A122382 A122383 A122384 this_sequence A122386 A122387 A122388
%Y A122385 Sequence in context: A126602 A024681 A007495 this_sequence A035002 A032578 A035659
%K A122385 nonn
%O A122385 1,2
%A A122385 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 01 2006
%I A035002
%S A035002 1,1,1,2,2,2,4,5,5,4,8,12,14,12,8,16,28,37,37,28,16,32,64,94,106,94,64,
%T A035002 32,64,144,232,289,289,232,144,64,128,320,560,760,838,760,560,320,128,
%U A035002 256,704,1328,1944,2329,2329,1944,1328,704,256,512,1536,3104,4864,6266
%N A035002 Square array a(m,n) read by antidiagonals, where a(m,n) = sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).
%C A035002 a(m,n) is the sum of all the entries above it plus the sum of all the entries to the left of it.
%C A035002 a(m,n) equals the number of ways to move a chess rook from the lower left corner to square (m,n), with the rook moving only up or right. - Francisco Santos (santosf(AT)unican.es), Oct 20 2005
%D A035002 C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
%F A035002 G.f. A(n; x) for n-th row satisfies A(n; x) = Sum_{k=1..n} (1+x^k)*A(n-k; x), A(0; x) = 1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 03 2002
%F A035002 a(m+1, n+1)=2a(m+1, n)+2a(m, n+1)-3a(m, n); a(n, 1)=a(1, n)= A011782(n) - Francisco Santos (santosf(AT)unican.es), Oct 20 2005
%e A035002 Table begins:
%e A035002 1 1 2 4 8 16 32 64 ...
%e A035002 1 2 5 12 28 64 144 320 ...
%e A035002 2 5 14 37 94 232 560 1328 ...
%e A035002 4 12 37 106 289 760 1944 4864 ...
%Y A035002 Cf. A035001, A051708.
%Y A035002 Row sums give A025192.
%Y A035002 Adjacent sequences: A034999 A035000 A035001 this_sequence A035003 A035004 A035005
%Y A035002 Sequence in context: A024681 A007495 A122385 this_sequence A032578 A035659 A008282
%K A035002 nonn,tabl,easy,nice
%O A035002 1,4
%A A035002 Erich Friedman (erich.friedman(AT)stetson.edu)
%I A032578
%S A032578 1,1,1,1,1,1,1,1,1,2,2,2,4,5,5,5,5,11,10,10,11,24,23,23,27,27,25,27,27,
%T A032578 27,62,55,53,63,335,310,300,326,713,703,690,726,704,755,762,1691,1611,
%U A032578 1587,1665,1574,1716,4122,3719,4097,4057,4005
%N A032578 Quotient of 'base 23' division described in A032577.
%Y A032578 Cf. A032577. See also A032563 for explanation.
%Y A032578 Adjacent sequences: A032575 A032576 A032577 this_sequence A032579 A032580 A032581
%Y A032578 Sequence in context: A007495 A122385 A035002 this_sequence A035659 A008282 A074765
%K A032578 nonn,hard
%O A032578 0,10
%A A032578 Patrick De Geest (pdg(AT)worldofnumbers.com), april 1998.
%E A032578 More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000
%E A032578 The next term has an A032577 value > 2.4*10^11
%I A035659
%S A035659 0,0,0,0,1,1,1,1,2,2,2,4,5,5,5,7,8,8,11,14,15,15,19,22,23,27,33,37,38,
%T A035659 44,50,54,60,71,79,84,94,106,114,125,143,159,169,187,208,225,242,273,
%U A035659 302,323,351,390,421,451,498,551,590,636,697,757,807,881,965,1039
%N A035659 Number of partitions of n into parts 7k+1 and 7k+4 with at least one part of each type.
%Y A035659 Adjacent sequences: A035656 A035657 A035658 this_sequence A035660 A035661 A035662
%Y A035659 Sequence in context: A122385 A035002 A032578 this_sequence A008282 A074765 A029045
%K A035659 nonn
%O A035659 1,9
%A A035659 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A008282
%S A008282 1,1,1,1,2,2,2,4,5,5,5,10,14,16,16,16,32,46,56,61,61,61,122,
%T A008282 178,224,256,272,272,272,544,800,1024,1202,1324,1385,1385,
%U A008282 1385,2770,4094,5296,6320,7120,7664,7936,7936,7936,15872
%N A008282 Triangle of Euler-Bernoulli or Entringer numbers read by rows: T(n,k) is the number of down-up permutations of n+1 starting with k+1.
%C A008282 Triangle begins
%C A008282 1
%C A008282 1 1
%C A008282 1 2 2
%C A008282 2 4 5 5
%C A008282 5 10 14 16 16
%C A008282 16 32 46 56 61 61
%C A008282 ...
%C A008282 Each row is constructed by forming the partial sums of the previous row, reading from the right, and repeating the final term.
%D A008282 V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
%D A008282 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A008282 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
%D A008282 C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
%H A008282 J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
%H A008282 B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A008282 B. Gourevitch, L'univers de Pi
%H A008282 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
%F A008282 T(n, k)=sum((-1)^i*binomial(k, 2i+1)*E[n-2i-1], i=0..floor((k-1)/2))= sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (kChallenge Problems: Independent Sets in Graphs
%H A057591 N. J. A. Sloane, On single-deletion-correcting codes
%Y A057591 Cf. A000016, A057608, A057657, A010101.
%Y A057591 Adjacent sequences: A057588 A057589 A057590 this_sequence A057592 A057593 A057594
%Y A057591 Sequence in context: A008282 A074765 A029045 this_sequence A024405 A082547 A068928
%K A057591 nice,hard,nonn
%O A057591 1,3
%A A057591 njas, Oct 05 2000
%E A057591 Guenter Stertenbrink (Sterten(AT)aol.com) found a(9) = 11 and a(10) >= 16, Apr 28, 2001.
%E A057591 James B. Shearer (jbs(AT)pkmfgvm4.vnet.ibm.com) proved that a(10 = 16, Sep 20, 2003.
%I A024405
%S A024405 1,1,2,2,2,4,5,7,13,19,36,56,102,193,350,642,1131,2191,3975,7480,14540,
%T A024405 27004,51067,95327,174777,335535,658017,1266162,2485494,4793475,8521778,
%U A024405 16519017,31580551,62245705,116080359,229064578,440510030,848388496
%N A024405 Number of products of distinct primes <= p(n) equal to 1 (mod p(n)).
%Y A024405 Adjacent sequences: A024402 A024403 A024404 this_sequence A024406 A024407 A024408
%Y A024405 Sequence in context: A074765 A029045 A057591 this_sequence A082547 A068928 A086420
%K A024405 nonn
%O A024405 1,3
%A A024405 David W. Wilson (davidwwilson(AT)comcast.net)
%I A082547
%S A082547 1,2,2,2,4,5,9,11,16,22,26,30,38,45,51,59,68,77,83,96,106,115,127,139,
%T A082547 151,165,177,190,204,221,236,250,267,286,304,323,339,361,382,400,421,
%U A082547 440,465,486,512,533,556,580,604,633,656,686,713,739,769,797,827,856
%N A082547 Number of primes p such that p can be expressed as the sum of distinct primes with largest prime in the sum = n-th prime.
%e A082547 For n=5; 11 is the 5th prime. 11=11, 13= 2+11, 19= 3+5+11, 23= 2+3+7+11 = 5+7+11. 11 and 13,19,23 are primes. so a(5)=4.
%o A082547 (PARI) limit = 70; M = sum(i = 1, limit, prime(i)); v = vector(M); primeSum = 0; forprime (n = 1, prime(limit), count = 1; forstep (i = primeSum, 1, -1, if (v[i], if (isprime(i + n), count = count + 1); v[i + n] = 1)); v[n] = 1; print(count); primeSum = primeSum + n)
%Y A082547 Cf. A082533, A082534.
%Y A082547 Adjacent sequences: A082544 A082545 A082546 this_sequence A082548 A082549 A082550
%Y A082547 Sequence in context: A029045 A057591 A024405 this_sequence A068928 A086420 A103265
%K A082547 easy,nonn
%O A082547 1,2
%A A082547 Naohiro Nomoto (n_nomoto(AT)yabumi.com), May 02 2003
%E A082547 More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Sep 16 2004
%I A068928
%S A068928 2,2,2,4,5,9,12,21,30,51,76,127,195,322,504,826,1309,2135,3410,5545,
%T A068928 8900,14445,23256,37701,60813,98514,159094,257608,416325,673933,
%U A068928 1089648,1763581,2852242,4615823,7466468,12082291,19546175,31628466
%N A068928 Number of incongruent ways to tile a 3 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
%F A068928 For n >= 8, a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-5) - a(n-6).
%Y A068928 Cf. A068922 for total number of tilings, A068926 for more info.
%Y A068928 Essentially the same as A001224.
%Y A068928 Adjacent sequences: A068925 A068926 A068927 this_sequence A068929 A068930 A068931
%Y A068928 Sequence in context: A057591 A024405 A082547 this_sequence A086420 A103265 A008238
%K A068928 easy,nonn
%O A068928 1,1
%A A068928 Dean Hickerson (dean(AT)math.ucdavis.edu), Mar 11 2002
%I A086420
%S A086420 1,1,2,2,2,4,6,4,8,6,8,18,16,12,16,18,32,24,54,32,36,64,48,54,64,72,162,
%T A086420 128,96,108,128,144,162,256,192,216,486,256,288,324,512,384,432,486,512,
%U A086420 576,648,1024,1458,768,864,972,1024,1152,1296,2048,1458,1536
%N A086420 Euler's totient of 3-smooth numbers: a(n)=A000010(A003586(n)).
%C A086420 a(n) is 3-smooth.
%H A086420 Eric Weisstein's World of Mathematics, Totient Function.
%H A086420 Eric Weisstein's World of Mathematics, Smooth Number
%F A086420 n>1: a(n) = A003586(n) * (if A003586(n) mod 3 > 0 then 1/2 else (1 + A003586(n) mod 2)/3), a(1)=1.
%Y A086420 Adjacent sequences: A086417 A086418 A086419 this_sequence A086421 A086422 A086423
%Y A086420 Sequence in context: A024405 A082547 A068928 this_sequence A103265 A008238 A096575
%K A086420 nonn
%O A086420 1,3
%A A086420 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 18 2003
%I A103265
%S A103265 1,2,2,2,4,6,6,6,8,12,14,14,16,22,26,26,30,38,44,46,52,62,70,74,80,96,
%T A103265 110,116,124,146,166,174,186,210,238,254,272,302,338,362,384,426,470,
%U A103265 502,532,588,646,686,726,792,872,926,980,1062
%N A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There no restriction on parts which are twice squares.
%F A103265 Gf: product_{k>0}((1+x^k^2)/(1-x^k^2)).
%e A103265 E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.
%p A103265 series(product((1+x^(k^2))/(1-x^(k^2)),k=1..100),x=0,100);
%Y A103265 Adjacent sequences: A103262 A103263 A103264 this_sequence A103266 A103267 A103268
%Y A103265 Sequence in context: A082547 A068928 A086420 this_sequence A008238 A096575 A002722
%K A103265 easy,nonn
%O A103265 0,2
%A A103265 Noureddine Chair (n.chair(AT)rocketmail.com.com), Feb 27 2005
%I A008238
%S A008238 0,0,0,0,1,2,2,2,4,6,6,6,9,12,12,12,16,20,20,20,25,30,30,
%T A008238 30,36,42,42,42,49,56,56,56,64,72,72,72,81,90,90,90,100,
%U A008238 110,110,110,121,132,132,132,144,156,156,156,169,182,182
%N A008238 floor(n/4)*ceil(n/4).
%Y A008238 Adjacent sequences: A008235 A008236 A008237 this_sequence A008239 A008240 A008241
%Y A008238 Sequence in context: A068928 A086420 A103265 this_sequence A096575 A002722 A093393
%K A008238 nonn
%O A008238 0,6
%A A008238 njas
%I A096575
%S A096575 1,1,1,2,2,2,4,6,6,8,11,13,17,24,28
%N A096575 Number of fixed points of solid partitions under rotation operation.
%C A096575 Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link above.
%H A096575 Wouter Meeussen, Solid Partitions Mma functions
%e A096575 Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.
%t A096575 turn[par_List] := Module[{maks, wide, it}, wide = Length[par[[1]]]; maks = Max[Length[par], wide, Flatten[par]]; it = Join[ #, Table[0, {wide - Length[ # ]}]] & /@( par /. i_Integer :> Table[If[w > i, 0, 1], {w, maks}]); DeleteCases[DeleteCases[Transpose[Apply[Plus, it, 1]], 0 | {}, -1], 0|{}, -1]]; Table[sn =Sort@Flatten[solidformBTK /@ Partitions[n]]; Frequencies[Length /@ ToCycles[Ordering[Map[turn @ # &, sn, {2}]]] ], {n, 1, 15}]
%Y A096575 Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096576, A096577, A096578, A096579, A096580, A096581.
%Y A096575 Adjacent sequences: A096572 A096573 A096574 this_sequence A096576 A096577 A096578
%Y A096575 Sequence in context: A086420 A103265 A008238 this_sequence A002722 A093393 A090858
%K A096575 more,nonn
%O A096575 1,4
%A A096575 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jun 27 2004
%I A002722 M0238 N0081
%S A002722 1,1,1,2,2,2,4,6,6,8,11,13,17,24,28,36
%N A002722 Rotatable partitions.
%D A002722 E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.
%Y A002722 Adjacent sequences: A002719 A002720 A002721 this_sequence A002723 A002724 A002725
%Y A002722 Sequence in context: A103265 A008238 A096575 this_sequence A093393 A090858 A036654
%K A002722 nonn
%O A002722 1,4
%A A002722 njas
%I A093393
%S A093393 0,0,0,1,2,2,2,4,6,7,7,8,9,9,9,10,12,13,14,15,16,16,16,17,18,19,20,22,
%T A093393 23,23,23,24,25,25,26,28,30,30,30,31,32,32,32,34,36,37,37,38,39,39,39,
%U A093393 40,42,43,44,45,46,46,46,47,48,49,50,52,53,53,53,54,55,55,56,58,60,60
%N A093393 [n/9] + [n/4] + [(n+1)/9] + [(n+1)/4] + [(n+2)/9].
%C A093393 a(n) = A004524(n) + A093390(n).
%Y A093393 Adjacent sequences: A093390 A093391 A093392 this_sequence A093394 A093395 A093396
%Y A093393 Sequence in context: A008238 A096575 A002722 this_sequence A090858 A036654 A010558
%K A093393 nonn
%O A093393 0,5
%A A093393 Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Mar 28 2004
%I A090858
%S A090858 0,1,0,2,2,2,4,6,7,8,13,15,21,25,30,39,50,58,74,89,105,129,156,185,221,
%T A090858 264,309,366,433,505,593,696,805,941,1090,1258,1458,1684,1933,2225,2555,
%U A090858 2922,3346,3823,4349,4961,5644,6402,7267,8234,9309,10525,11886,13393
%N A090858 Number of partitions of n such that there is only one part which occurs twice, while all other parts occur only once.
%C A090858 Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.
%C A090858 Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1],[3,3,1],[3,2,2], and[3,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%F A090858 G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
%F A090858 G.f.=sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%e A090858 a(7)= 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
%p A090858 g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
%Y A090858 Cf. A047967.
%Y A090858 Adjacent sequences: A090855 A090856 A090857 this_sequence A090859 A090860 A090861
%Y A090858 Sequence in context: A096575 A002722 A093393 this_sequence A036654 A010558 A060827
%K A090858 easy,nonn
%O A090858 1,4
%A A090858 Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 12 2004
%E A090858 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
%I A036654
%S A036654 1,1,1,2,2,2,4,6,8,8,10,18,34,52,52,24,58,136,288,472,472,66,186,
%T A036654 538,1424,3224,5504,5504,180,614,2080,6648,18888,44712,78416,78616,
%U A036654 522,2034,7970,29700,101340,302096,738448,1320064,1320064
%N A036654 Triangle of series-parallel numbers.
%D A036654 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
%e A036654 1; 1 1; 2 2 2; 4 6 8 8;...
%Y A036654 Adjacent sequences: A036651 A036652 A036653 this_sequence A036655 A036656 A036657
%Y A036654 Sequence in context: A002722 A093393 A090858 this_sequence A010558 A060827 A064355
%K A036654 nonn,easy,tabl,nice
%O A036654 0,4
%A A036654 njas
%I A010558
%S A010558 1,1,1,2,2,2,4,6,10
%N A010558 Maximal size of binary code of length n correcting 2 unidirectional errors.
%D A010558 J. H. Weber, Bounds and Constructions for Binary Block Codes Correcting Asymmetric or Unidirectional Errors, Ph. D. Thesis, Tech. Univ. Delft, 1989.
%D A010558 J. H. Weber, C. de Vroedt and D. E. Boekee, Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6, IEEE Trans. Inform. Theory, 34 (1988), 1321-1332.
%Y A010558 Adjacent sequences: A010555 A010556 A010557 this_sequence A010559 A010560 A010561
%Y A010558 Sequence in context: A093393 A090858 A036654 this_sequence A060827 A064355 A000799
%K A010558 nonn
%O A010558 1,4
%A A010558 njas
%I A060827
%S A060827 2,2,2,4,6,10,12,22,28,50,62,112,140,252,314,566,706,1272,1586,2858,
%T A060827 3564,6422,8008,14430,17994,32424,40432,72856,90850,163706,204138,
%U A060827 367844,458694,826538,1030676,1857214,2315908,4173122,5203798,9376920
%N A060827 3-wave sequence beginning with 2's.
%C A060827 The 3-wave sequence with initial values a, b, c is formed by the following construction:
%C A060827 a.......a+b+c............3a+5b+6c...
%C A060827 ..b...b+c...a+2b+2c..2a+4b+5c...
%C A060827 ....c..........a+2b+3c...
%C A060827 Dropping middle row gives A052994.
%H A060827 F. v. Lamoen, Wave sequences
%F A060827 G.f.: (-2x^2+2x+2)/(x^6-x^4-2x^2+1).
%Y A060827 Cf. A038196.
%Y A060827 Adjacent sequences: A060824 A060825 A060826 this_sequence A060828 A060829 A060830
%Y A060827 Sequence in context: A090858 A036654 A010558 this_sequence A064355 A000799 A063823
%K A060827 easy,nonn
%O A060827 0,1
%A A060827 Jason Earls (jcearls(AT)cableone.net), Apr 30 2001
%E A060827 More terms from Larry Reeves (larryr(AT)acm.org), May 10 2001
%I A064355
%S A064355 2,2,2,4,6,10,18,32,56,102,186,340,630,1170,2182,4096,7710,14560,27594,
%T A064355 52428,99858,190650,364722,699040,1342176,2581110,4971008,9586980,
%U A064355 18512790,35791358,69273666,134217728,260300986,505290270,981706806
%N A064355 Number of subsets of {1,2,..n} which sum to 1 mod n.
%H A064355 Index entries for sequences related to subset sums modulo m
%F A064355 a(n) = 1/n * sum_{d divides n and d is odd} 2^(n/d) * mu(d); (mu(d) is the Moebius function, sequence A008683).
%e A064355 a(7)=18 because there are 18 subsets of {1,2,3,4,5,6,7} which sum to 1 mod 7:{1}, {1,7}, {2,6}, {3,5}, {1,2,5}, {1,3,4}, {2,6,7}, {3,5,7}, {4,5,6}, {1,2,5,7}, {1,3,4,7}, {1,3,5,6}, {2,3,4,6}, {4,5,6,7}, {1,2,3,4,5}, {1,3,5,6,7}, {2,3,4,6,7}, {1,2,3,4,5,7}.
%t A064355 f[n_] := Block[{d = Select[Divisors@n, OddQ@ # &]}, Plus @@ (2^(n/d)*MoebiusMu@d)/n]; Array[f, 35] (from Robert G. Wilson v (rgwv(at)rgwv.com), Feb 20 2006)
%Y A064355 Cf. A063776, A008683. Equals 2*A000048(n).
%Y A064355 Adjacent sequences: A064352 A064353 A064354 this_sequence A064356 A064357 A064358
%Y A064355 Sequence in context: A036654 A010558 A060827 this_sequence A000799 A063823 A005865
%K A064355 nonn
%O A064355 1,1
%A A064355 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Sep 25 2001
%E A064355 More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 27 2001
%I A000799 M0239 N0082
%S A000799 2,2,2,4,6,10,18,32,56,102,186,341,630,1170,2184,4096,7710,14563,
%T A000799 27594,52428,99864,190650,364722,699050,1342177,2581110,4971026,
%U A000799 9586980,18512790,35791394,69273666,134217728,260301048,505290270
%N A000799 Floor( 2^n /n ).
%Y A000799 Cf. A000801.
%Y A000799 Cf. A065482, A082482, A053638.
%Y A000799 Adjacent sequences: A000796 A000797 A000798 this_sequence A000800 A000801 A000802
%Y A000799 Sequence in context: A010558 A060827 A064355 this_sequence A063823 A005865 A098705
%K A000799 nonn
%O A000799 1,1
%A A000799 njas
%I A063823
%S A063823 1,2,2,2,4,6,10,20,38,74,148,294,586,1172,2342,4682,9364,18726,
%T A063823 37450,74900,149798,299594,599188,1198374,2396746,4793492,9586982,
%U A063823 19173962,38347924,76695846,153391690,306783380,613566758,1227133514
%N A063823 G.f.: (1-2*x^2-3*x^3)/((1-x^3)*(1-2*x))
%D A063823 P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 259.
%Y A063823 Adjacent sequences: A063820 A063821 A063822 this_sequence A063824 A063825 A063826
%Y A063823 Sequence in context: A060827 A064355 A000799 this_sequence A005865 A098705 A029866
%K A063823 nonn
%O A063823 0,2
%A A063823 njas, Aug 21 2001
%I A005865 M0240
%S A005865 1,1,1,1,1,2,2,2,4,6,12,24,32,64,128,256
%N A005865 The coding-theoretic function A(n,6).
%C A005865 Since A(n,5) = A(n+1,6), A(n,5) gives essentially the same sequence.
%D A005865 A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H A005865 S. Litsyn, E. M. Rains and N. J. A. Sloane, A(n,d) tables.
%H A005865 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%H A005865 Index entries for sequences related to A(n,d)
%Y A005865 Cf. A005864, A005866.
%Y A005865 Adjacent sequences: A005862 A005863 A005864 this_sequence A005866 A005867 A005868
%Y A005865 Sequence in context: A064355 A000799 A063823 this_sequ