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

1,5

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.

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

Example

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

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]]

Crossrefs

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

Sequence in context: A318498 A093997 A157196 * A362845 A293451 A063014

Adjacent sequences: A300407 A300408 A300409 * A300411 A300412 A300413

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 05 2018

Status

approved