A300410 Number of centered square numbers dividing n.

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, 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, 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

Offset: 1

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

• nonn

Examples

			a(26) = 2 because 26 has 4 divisors {1, 2, 13, 26} among which 2 divisors {1, 13} are centered square numbers.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Centered Square Number.
• Index entries for sequences related to centered polygonal numbers.

Crossrefs

Cf. A001844, A046951, A228048, A279496, A300409.

Programs

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
for k from 1 do
  m:= 2*k*(k+1)+1;
  if m > N then break fi;
  r:= [seq(i,i=m..N,m)];
  V[r]:= map(t->t+1, V[r]);
od:
convert(V,list); # Robert Israel, Mar 05 2018

Mathematica

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]]

Formula

G.f.: Sum_{k>=0} x^(2*k*(k+1)+1)/(1 - x^(2*k*(k+1)+1)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A228048 = 1.440659... . - Amiram Eldar, Jan 02 2024