A244964 Number of distinct generalized pentagonal numbers dividing n.

1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 4, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 1, 3, 1, 3, 3, 2, 1, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 1, 5, 1, 2, 2, 2, 2, 3, 1, 2, 1, 6, 1, 3, 1, 2, 3, 2, 3, 3, 1, 4, 1, 2, 1, 4, 2, 2, 1, 3, 1, 4, 2, 3, 1, 2, 2, 3, 1, 3, 1, 4, 1, 3, 1, 3, 5

Offset

1,2

Comments

For more information about the generalized pentagonal numbers see A001318.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Dec 31 2023: (Start)

a(n) = Sum_{d|n} A080995(d).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

Example

For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

Mathematica

a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

Crossrefs

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

Sequence in context: A342510 A298071 A246920 * A096993 A297774 A043533

Adjacent sequences: A244961 A244962 A244963 * A244965 A244966 A244967

Keyword

nonn

Author

Omar E. Pol, Jul 10 2014

Status

approved