A280022 - OEIS

%I #36 Oct 31 2023 04:44:52

%S 0,1,48,324,1792,3750,15552,19208,61440,85293,180000,175692,580608,

%T 399854,921984,1215000,2031616,1503378,4094064,2606420,6720000,

%U 6223392,8433216,6716184,19906560,12109375,19192992,21257640,34420736,21218430,58320000,29552672

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

%C Multiplicative because A000203 is. - _Andrew Howroyd_, Jul 23 2018

%H Seiichi Manyama, <a href="/A280022/b280022.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

%F From _Amiram Eldar_, Oct 31 2023: (Start)

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

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

%t Table[n^4 * DivisorSigma[1, n], {n, 0, 32}] (* _Amiram Eldar_, Oct 31 2023 *)

%o (PARI) a(n) = if(n < 1, 0, n^4 * sigma(n)); \\ _Andrew Howroyd_, Jul 23 2018

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

%Y 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).

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

%Y Cf. A353908.

%K nonn,easy,mult

%O 0,3

%A _Seiichi Manyama_, Feb 22 2017