Search: id:A038548
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%I A038548
%S A038548 1,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,4,2,2,2,3,1,4,1,3,2,2,2,
%T A038548 5,1,2,2,4,1,4,1,3,3,2,1,5,2,3,2,3,1,4,2,4,2,2,1,6,1,2,3,4,2,4,1,3,2,4,
%U A038548 1,6,1,2,3,3,2,4,1,5,3,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,3,3,5,1,4,1,4,4
%N A038548 Number of divisors of n that are at most sqrt(n).
%C A038548 Number of ways to arrange n identical objects in a rectangle, modulo 
               rotation.
%C A038548 Number of unordered solutions of xy = n. - Colin Mallows (colinm(AT)research.avayalabs.com) 
               Jan 26 2002
%C A038548 Number of ways to write n-1 as n-1 = x*y + x + y, 0<=x<=y<=n. - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Jun 23 2002
%C A038548 Also number of values for x where x+2n and x-2n are both squares [e.g. 
               if n=9, then 18+18 and 18-18 are both squares, as are 82+18 and 82-18 
               so a(9)=2]; this is because a(n) is the number of solutions to n=k(k+r) 
               in which case if x=r^2+2n then x+2n=(r+2k)^2 and x-2n=r^2 (cf. A061408). 
               - Henry Bottomley (se16(AT)btinternet.com), May 03 2001
%C A038548 Also number of sums of sequences of consecutive odd numbers or consecutive 
               even numbers including sequences of length 1 (e.g. 12 = 5+7 or 2+4+6 
               or 12 so a(12)=3). - Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 
               26 2002
%C A038548 Number of partitions whose consecutive parts differ by exactly two.
%C A038548 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) 
               since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1) - Christian 
               G. Bower (bowerc(AT)usa.net), Jun 06 2005.
%C A038548 Also number of partitions of n such that if k is the largest part, then 
               each of the parts 1,2,...,k-1 occurs exactly twice. Example: a(12)=3 
               because we have [3,3,2,2,1,1],[2,2,2,2,2,1,1] and [1,1,1,1,1,1,1,
               1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 07 2006
%C A038548 a(n) is also the number of nonnegative integer solutions of the Diophantine 
               equation 4 x^2-y^2=16 n. For example a(24)=4 because there are 4 
               solutions :(x,y)=(10,4),(11,10),(14,20),(25,46). - Nour-Eddine Fahssi 
               (fahssin(AT)yahoo.fr), Feb 27 2008
%C A038548 Contribution from Eric Desbiaux (moongerms(AT)wanadoo.fr), Nov 16 2008: 
               (Start)
%C A038548 Sum(n=1...inf) ((10^(-(n)))^(n))*(10^n)/(10^n-1)
%C A038548 =Sum(n=1...inf)((10^(-n))^((-1)^n))*(1/(10^n-1))
%C A038548 =Sum(n=1...inf)((10^(-n))^((-1)^n))*A73668 = A038548
%C A038548 (End)
%D A038548 G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 
               2004. page 18 Exer. 21,22
%H A038548 T. D. Noe, <a href="b038548.txt">Table of n, a(n) for n=1..10000</a>
%H A038548 T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, 
               Vol. 2, 1999, #99.1.6.
%F A038548 ceiling(d(n)/2), where d(n) = number of divisors of n (A000005)
%F A038548 a(2k) = A034178(2k)+A001227(k). a(2k+1) = A034178(2k+1). - Naohiro Nomoto 
               (n_nomoto(AT)yabumi.com), Feb 26 2002
%F A038548 G.f.: sum(k=1, oo, x^(k^2)/(1-x^k)) - Jon Perry (perry(AT)globalnet.co.uk), 
               Sep 10 2004
%F A038548 Dirichlet g.f.: (zeta(s) + zeta(2*s))/2. - Christian G. Bower (bowerc(AT)usa.net), 
               Jun 06 2005.
%F A038548 a(n) = (A000005(n) + A010052(n))/2. [From Omar E. Pol (info(AT)polprimos.com), 
               Jun 23 2009]
%p A038548 with(numtheory): A038548 := n->ceil(sigma[0](n)/2);
%t A038548 Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] [From Robert G. 
               Wilson, v (rgwv(AT)rgwv.com), Mar 02 2009]
%o A038548 (PARI) a(n)=if(n>=0,sumdiv(n,d,d*d<=n)) /* Michael Somos Jan 25 2005 
               */
%Y A038548 Records give A038549, A004778, A086921. Cf. A000005, A072670.
%Y A038548 a(A025487) = A108504.
%Y A038548 Cf. A010052, A161841, A161842. [From Omar E. Pol (info(AT)polprimos.com), 
               Jun 23 2009]
%Y A038548 Sequence in context: A076755 A106490 A122375 this_sequence A068108 A113309 
               A062362
%Y A038548 Adjacent sequences: A038545 A038546 A038547 this_sequence A038549 A038550 
               A038551
%K A038548 nonn,easy,nice
%O A038548 1,4
%A A038548 Tom Verhoeff (Tom.Verhoeff(AT)acm.org), N. J. A. Sloane (njas(AT)research.att.com).

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