%I A046951
%S A046951 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,3,1,1,1,
%T A046951 4,1,1,1,2,1,1,1,2,2,1,1,3,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,4,1,1,1,2,1,1,
%U A046951 1,4,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,2,2,4,1,1,1,2,1
%N A046951 a(n) = |{(i,j):i*j=n AND i|j}| = |{(i,j):i*j^2=n}|. Also tau(A000188); 
               also number of squares dividing n.
%C A046951 Invented by the HR automatic theory formation program.
%C A046951 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).
%C A046951 a(A130279(n))=n and a(m)<>n for m<A130279(n); A008966(n)=0^(a(n)-1). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2007
%C A046951 sum_{k=1;inf} 1/(10^(k^2)-1) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), 
               Mar 24 2009]
%C A046951 We have a(n) = A159631(n) for all n<125, but a(125)=2<3=A159631(125). 
               [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 22 2009]
%H A046951 R. Zumkeller, <a href="b046951.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A046951 S. Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, 
               Vol. 2, 1999, #2.
%H A046951 S. Colton, <a href="http://www.dai.ed.ac.uk/homes/simonco/research/hr/
               ">HR - Automatic Theory Formation in Pure Mathematics</a>
%H A046951 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A046951 Multiplicative with p^e --> floor(e/2)+1, p prime. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), May 20 2007
%F A046951 Inverse Moebius transform of characteristic function of squares. Dirichlet 
               g.f.: zeta(s)*zeta(2s).
%F A046951 First differences of A013936. Average value tends towards pi^2/6=1.644934... 
               (A013661, A013679). - Henry Bottomley (se16(AT)btinternet.com), Aug 
               16 2001
%F A046951 G.f.: Sum_{k>0} x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Dec 13 2002
%e A046951 f(16) = 3 because 1*16=16 and 1|16, 2*8=16 and 2|8, 4*4=16 and 4|4.
%Y A046951 Cf. A000188, A004101, A005117, A038538, A046952, A052304. a(p^k)=A008619=[n/
               2]+1. a(A002110)=1.
%Y A046951 Cf. A159631 [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 22 2009]
%Y A046951 Sequence in context: A088737 A096309 A049419 this_sequence A159631 A050377 
               A001826
%Y A046951 Adjacent sequences: A046948 A046949 A046950 this_sequence A046952 A046953 
               A046954
%K A046951 nice,nonn,mult
%O A046951 1,4
%A A046951 Simon Colton (simonco(AT)cs.york.ac.uk)