A116913 Inverse Moebius transform of pentagonal numbers.

1, 6, 13, 28, 36, 69, 71, 120, 130, 186, 177, 301, 248, 363, 378, 496, 426, 663, 533, 798, 734, 897, 783, 1245, 961, 1254, 1210, 1547, 1248, 1914, 1427, 2016, 1806, 2148, 1926, 2821, 2036, 2685, 2522, 3270, 2502, 3702, 2753, 3801, 3510, 3939, 3291, 5053, 3648

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} d*(3*d-1)/2.

G.f.: Sum_{k>=1} k*(3*k-1)/2*x^k/(1 - x^k). - Ilya Gutkovskiy, May 23 2017

From Amiram Eldar, Dec 29 2024: (Start)

a(n) = (3*sigma_2(n) - sigma(n)) / 2 = (3*A001157(n) - A000203(n)) / 2.

Dirichlet g.f.: zeta(s) * (3*zeta(s-2) - zeta(s-1))/2.

Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)

Mathematica

Table[Sum[d*(3d - 1)/2, {d, Divisors[n]}], {n, 101}] (* Indranil Ghosh, May 23 2017 *)

Prog

(PARI) a(n) = sumdiv(n, d, d*(3*d-1)/2); \\ Michel Marcus, Mar 25 2015
(PARI) a(n) = {my(f = factor(n)); (3 * sigma(f, 2) - sigma(f)) / 2; } \\ Amiram Eldar, Dec 29 2024

Crossrefs

Cf. A000326 (pentagonal numbers), A000203, A002117.

Inverse Moebius transforms of polygonal numbers: A007437 (k=3), A001157 (k=4), this sequence (k=5), A278945 (k=6), A278947 (k=7).

Sequence in context: A301687 A173559 A086224 * A016071 A086652 A159694

Adjacent sequences: A116910 A116911 A116912 * A116914 A116915 A116916

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 19 2006

Extensions

More terms from Michel Marcus, Mar 25 2015

Status

approved