A280022 Expansion of phi_{5, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 48, 324, 1792, 3750, 15552, 19208, 61440, 85293, 180000, 175692, 580608, 399854, 921984, 1215000, 2031616, 1503378, 4094064, 2606420, 6720000, 6223392, 8433216, 6716184, 19906560, 12109375, 19192992, 21257640, 34420736, 21218430, 58320000, 29552672

Offset

0,3

Comments

Multiplicative because A000203 is. - Andrew Howroyd, Jul 23 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = n^4*A000203(n) for n > 0.

a(n) = (15*A282101(n) - 20*A282595(n) + 10*A282586(n) - 4*A013974(n) - A282431(n))/20736.

Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^2/36 = 0.274155... (A353908). - Amiram Eldar, Dec 08 2022

From Amiram Eldar, Oct 31 2023: (Start)

Multiplicative with a(p^e) = p^(4*e) * (p^(e+1)-1)/(p-1).

Dirichlet g.f.: zeta(s-4)*zeta(s-5). (End)

Mathematica

Table[n^4 * DivisorSigma[1, n], {n, 0, 32}] (* Amiram Eldar, Oct 31 2023 *)

Prog

(PARI) a(n) = if(n < 1, 0, n^4 * sigma(n)); \\ Andrew Howroyd, Jul 23 2018

Crossrefs

Cf. this sequence (phi_{5, 4}), A280025 (phi_{7, 4}).

Cf. A282101 (E_2*E_4^2), A282595 (E_2^2*E_6), A282586 (E_2^3*E_4), A013974 (E_4*E_6 = E_10), A282431 (E_2^5).

Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), A282097 (n^2*sigma(n)), A282211 (n^3*sigma(n)), this sequence (n^4*sigma(n)).

Cf. A353908.

Sequence in context: A134607 A192215 A347613 * A223396 A211735 A211746

Adjacent sequences: A280019 A280020 A280021 * A280023 A280024 A280025

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 22 2017

Status

approved