A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

0, 1, 144, 2268, 18688, 78750, 326592, 825944, 2396160, 4966677, 11340000, 19501812, 42384384, 62777078, 118935936, 178605000, 306774016, 410422194, 715201488, 894002060, 1471680000, 1873240992, 2808260928, 3405105288, 5434490880, 6152734375, 9039899232

Offset

0,3

Comments

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Links

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

Formula

a(n) = n^4*A001158(n) for n > 0.

a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.

Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - Amiram Eldar, Dec 08 2022

From Amiram Eldar, Oct 31 2023: (Start)

Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).

Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)

Mathematica

Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* Amiram Eldar, Oct 31 2023 *)

Prog

(PARI) a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ Andrew Howroyd, Jul 23 2018

Crossrefs

Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).

Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).

Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).

Cf. A152649.

Sequence in context: A066445 A008431 A250084 * A223687 A268625 A121344

Adjacent sequences: A280022 A280023 A280024 * A280026 A280027 A280028

Keyword

nonn,easy,mult

Author

Seiichi Manyama, Feb 22 2017

Status

approved