A279497 Number of pentagonal numbers dividing n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3

Offset

1,5

Links

Table of n, a(n) for n=1..120.

Eric Weisstein's World of Mathematics, Pentagonal Number.

Index to sequences related to polygonal numbers.

Formula

G.f.: Sum_{k>=1} x^(k*(3*k-1)/2)/(1 - x^(k*(3*k-1)/2)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*log(3) - Pi/sqrt(3) = 1.482037... (A244641). - Amiram Eldar, Jan 02 2024

Example

a(12) = 2 because 12 has 6 divisors {1,2,3,4,6,12} among which 2 divisors {1,12} are pentagonal numbers.

Mathematica

Rest[CoefficientList[Series[Sum[x^(k (3k -1)/2)/(1 - x^(k (3k -1)/2)), {k, 120}], {x, 0, 120}], x]]
Table[Count[Divisors[n], _?(IntegerQ[(1+Sqrt[1+24#])/6]&)], {n, 120}] (* Harvey P. Dale, Jan 05 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, ispolygonal(d, 5)); \\ Michel Marcus, Jul 27 2022

Crossrefs

Cf. A000326, A007862, A046951, A244641.

Sequence in context: A353445 A115722 A115721 * A359305 A138330 A128591

Adjacent sequences: A279494 A279495 A279496 * A279498 A279499 A279500

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 13 2016

Status

approved