1,1

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Cf. A006753.

Sequence in context: A088820 A058097 A131878 this_sequence A044124 A044505 A156797

Adjacent sequences: A019503 A019504 A019505 this_sequence A019507 A019508 A019509

nonn,base

Mario Velucchi (mathchess(AT)velucchi.it)

1,1

Two-sided primes: deleting any number of digits at left or at right, but not both, leaves a prime.

Primes in which every digit string containing the most significant digit or the least significant digit is prime. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 24 2003

Angell, I. O. and Godwin, H. J. "On Truncatable Primes." Math. Comput. 31, 265-267, 1977.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 178 (Rev. ed. 1997).

P. De Geest, The list of 4260 left-truncatable primes

Index entries for sequences related to truncatable primes

tspQ[n_] := Module[{idn=IntegerDigits[n], l}, l=Length[idn]; Union[PrimeQ/@(FromDigits/@ Join[Table[Take[idn, i], {i, l}], Table[Take[idn, -i], {i, l}]])]=={True}] Select[Prime[Range[PrimePi[740000]]], tspQ]

Cf. A033664, A024785, A032437, A024770, A052023, A052024, A052025, A050986, A050987.

Sequence in context: A104179 A096148 A124674 this_sequence A085823 A100552 A155873

Adjacent sequences: A020991 A020992 A020993 this_sequence A020995 A020996 A020997

nonn,fini,full,base

Mario Velucchi (mathchess(AT)velucchi.it)

Corrected by David W. Wilson.

Additional comments from Harvey P. Dale (hpd1(AT)nyu.edu), Jul 10 2002

0,2

Members of Diophantine pairs.

Solution to b(b-1) = 2a(a-1) in natural numbers; a = a(n), b = b(n) = A046090(n).

Mario Velucchi "The Pell's equation ... an amusing application" in Mathematics and Informatics Quarterly, to appear 1997.

a(n)= (A001653(n)+1)/2.

a(n)=(((1+Sqrt(2))^(2*n-1)-(1-Sqrt(2))^(2*n-1))/Sqrt(8)+1)/2.

a_n = 7[a_(n-1) - a_(n-2)] + a_(n-3); a_(1) = 1, a_(2) = 3, a_(3) = 15. Also a(n) = 1/2 + ( (1-sqrt(2))/(-4*sqrt(2)) )*(3-2*sqrt(2))^n + ( (1+sqrt(2))/(4*sqrt(2)) )*(3+2*sqrt(2))^n. - Antonio Alberto Olivares, Dec 23 2003

Sqrt(2) = Sum_{n=0..inf} 1/a(n); a(n)=a(n-1)+floor(1/(Sqrt(2)-Sum_{k=0..n-1}1/a(k))) (n>0) with a(0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 25 2004

For n>k, a(n+k)=A001541(n)*A001653(k)-A053141(n-k-1); e.g. 493=99*5-2. For n<=k, a(n+k)=A001541(n)*A001653(k)-A053141(k-n); e.g. 85=3*29-2 - Charlie Marion (charliemath(AT)optonline.net), Oct 18 2004

a(n+1)=3*a(n)-1+(8*a(n)^2-8*a(n)+1)^0.5, a(1)=1. - Richard Choulet (richardchoulet(AT)yahoo.fr), Sep 18 2007

G.f.: (1-4*x+x^2)/((1-x)*(1-6*x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2009]

Cf. A001653, A046090.

Sequence in context: A005809 A067122 A093593 this_sequence A118342 A084209 A127085

Adjacent sequences: A011897 A011898 A011899 this_sequence A011901 A011902 A011903

nonn,easy,new

Mario Velucchi (mathchess(AT)velucchi.it)

More terms and comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

1,2

T. D. Noe, Table of n, a(n) for n=1..100

Hisanori Mishima, Factorizations of many number sequences

R. G. Wilson v, Explicit factorizations

Table[Product[Prime[k], {k, 1, n}] - 1, {n, 1, 18}] - Artur Jasinski (grafix(AT)csl.pl), Jan 01 2007

Cf. A002110, A006862.

Sequence in context: A004213 A105277 A103213 this_sequence A030522 A091124 A121143

Adjacent sequences: A057585 A057586 A057587 this_sequence A057589 A057590 A057591

nonn,easy,nice

Mario Velucchi (mathchess(AT)velucchi.it), Oct 05 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

0,2

Mario Velucchi, Seeing couples, in Recreational and Educational Computing, to appear 1997.

sqrt 3 = 1 + Sum(1 through infinity) 2/a(n) = 2/2 + 2/3 + 2/33 + 2/451 + 2/6273 + 2/87363 + 2/1216801... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 12 2003

a(n)^2 = A103974(n+1)^2 - (4*A007655(n+1))^2. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 06 2005

Cf. A011916, A011918, A011920.

Cf. A103974, A007655.

Sequence in context: A163476 A155660 A009502 this_sequence A071405 A092170 A083080

Adjacent sequences: A011919 A011920 A011921 this_sequence A011923 A011924 A011925

nonn,easy

Mario Velucchi (mathchess(AT)velucchi.it)

Formula corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001

1,1

Primes whose reciprocals have even period length.

A002371(A049084(a(n))) mod 2 == 0.

Not the same as A040121: a(33)=241 is not in A040121.

Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 05 2008: (Start)

Let (d(i): 1<=i<=2*K) be the period of decimal expansion of 1/a(n), K=A002371(A049084(a(n)))/2,

then d(i) + d(i+K) = 9 for i with 1<=i<=K, or, equivalently:

u + v = 10^K - 1 with u=SUM(d(i)*10^(K-i):1<=i<=K) and v=SUM(d(i+K)*10^(K-i):1<=i<=K). (End)

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 05 2008]

Index entries for sequences related to decimal expansion of 1/n.

Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 05 2008: (Start)

(0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),

K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,

u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)

Cf. A087000.

Sequence in context: A135776 A067831 A086998 this_sequence A040121 A156114 A091554

Adjacent sequences: A028413 A028414 A028415 this_sequence A028417 A028418 A028419

nonn,base

Mario Velucchi (mathchess(AT)velucchi.it)

More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 29 2003

1,1

Integers n such that n^2 = sum(n+1,n+2,n+3,...,n+x) for some value of x. 3 is a term because 3^2=9 and 4+5=9; 44 is a term because 44^2=1936 and the sum of (45,46,47,...,76) = 1936. [From Gil Broussard (gilbroussard(AT)bellsouth.net), Dec 23 2008]

Mario VELUCCHI "Seeing couples" in Recreational and Educational Computing, to appear 1997.

Sequence in context: A055539 A046946 A092545 this_sequence A102811 A142600 A103980

Adjacent sequences: A011913 A011914 A011915 this_sequence A011917 A011918 A011919

nonn,easy

Mario Velucchi (mathchess(AT)velucchi.it)

1,2

Number of n X n binary matrices with n ones under action of dihedral group of the square D_4.

Mario Velucchi, Title?

Mario Velucchi, Different Dispositions in the ChessBoard.

See Velucchi link or the PARI program. Note that the polynomial whose coefficient of a^k is divided by 8 differs based upon whether the term's index is even or odd.

Let A(n) = C(n^2, n); B(n) = C((n^2-(n mod 2))/2, n/2); C(n) = C((n^2-(n mod 2))/4, n/4); D(n) = Sum(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p)). Then a(n) = (A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4), (A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4), (A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4), (A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4). - David W. Wilson (davidwwilson(AT)comcast.net), May 29 2003

For n=3 the 16 solutions are

111 110 110 110 110 110 110 101 101 101 100 100 100 010 010 010

000 100 010 001 000 000 000 010 000 000 011 010 001 110 101 010

000 000 000 000 100 010 001 000 100 010 000 001 010 000 000 010

(PARI) {p(a, b, N) = if(N%2==0, (a+b)^(N^2) + 2*(a+b)^N*(a^2+b^2)^((N^2-N)/2) + 3*(a^2+b^2)^(N^2/2) + 2*(a^4+b^4)^(N^2/4), (a+b)^(N^2) + 2*(a+b)*(a^4+b^4)^((N^2-1)/4) + (a+b)*(a^2+b^2)^((N^2-1)/2) + 4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))} for(k=1, 20, print1(polcoeff(p(a, 1, k), k)/8, ", "))

Cf. A054252 and A014409.

Sequence in context: A138764 A009833 A009044 this_sequence A090727 A108242 A140307

Adjacent sequences: A019315 A019316 A019317 this_sequence A019319 A019320 A019321

nonn,nice

Mario Velucchi (mathchess(AT)velucchi.it)

More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com) and David W. Wilson (davidwwilson(AT)comcast.net), May 28 2003

1,1

Mario VELUCCHI "Seeing couples" in Recreational and Educational Computing, to appear 1997.

Sequence in context: A034688 A132855 A051481 this_sequence A136300 A144997 A088756

Adjacent sequences: A011915 A011916 A011917 this_sequence A011919 A011920 A011921

nonn,easy

Mario Velucchi (mathchess(AT)velucchi.it)

1,5

Source

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Cf. A019317.

Sequence in context: A038944 A124081 A119573 this_sequence A093329 A111052 A138536

Adjacent sequences: A001363 A001364 A001365 this_sequence A001367 A001368 A001369

nonn

Mario Velucchi (mathchess(AT)velucchi.it)