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A300410 Number of centered square numbers dividing n.

%I A300410 #13 Feb 16 2025 08:33:53
%S A300410 1,1,1,1,2,1,1,1,1,2,1,1,2,1,2,1,1,1,1,2,1,1,1,1,3,2,1,1,1,2,1,1,1,1,
%T A300410 2,1,1,1,2,2,2,1,1,1,2,1,1,1,1,3,1,2,1,1,2,1,1,1,1,2,2,1,1,1,3,1,1,1,
%U A300410 1,2,1,1,1,1,3,1,1,2,1,2,1,2,1,1,3,1,1,1,1,2,2,1,1,1,2,1,1,1,1,3
%N A300410 Number of centered square numbers dividing n.
%H A300410 Robert Israel, <a href="/A300410/b300410.txt">Table of n, a(n) for n = 1..10000</a>
%H A300410 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>.
%H A300410 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%F A300410 G.f.: Sum_{k>=0} x^(2*k*(k+1)+1)/(1 - x^(2*k*(k+1)+1)).
%F A300410 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A228048 = 1.440659... . - _Amiram Eldar_, Jan 02 2024
%e A300410 a(26) = 2 because 26 has 4 divisors {1, 2, 13, 26} among which 2 divisors {1, 13} are centered square numbers.
%p A300410 N:= 100: # for a(1)..a(N)
%p A300410 V:= Vector(N,1):
%p A300410 for k from 1 do
%p A300410   m:= 2*k*(k+1)+1;
%p A300410   if m > N then break fi;
%p A300410   r:= [seq(i,i=m..N,m)];
%p A300410   V[r]:= map(t->t+1, V[r]);
%p A300410 od:
%p A300410 convert(V,list); # _Robert Israel_, Mar 05 2018
%t A300410 nmax = 100; Rest[CoefficientList[Series[Sum[x^(2 k (k + 1) + 1)/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]]
%Y A300410 Cf. A001844, A046951, A228048, A279496, A300409.
%K A300410 nonn
%O A300410 1,5
%A A300410 _Ilya Gutkovskiy_, Mar 05 2018