A288420 - OEIS

%I #24 Nov 14 2022 01:38:26

%S 1,17,85,273,631,1445,2409,4369,6898,10727,14653,23205,28575,40953,

%T 53635,69905,83539,117266,130341,172263,204765,249101,279865,371365,

%U 394406,485775,558778,657657,707311,911795,923553,1118481,1245505,1420163,1520079,1883154

%N a(n) = Sum_{d|n} d^4*A000593(n/d).

%C Multiplicative because this sequence is the Dirichlet convolution of A000583 and A000593 which are both multiplicative. - _Andrew Howroyd_, Jul 20 2018

%H Seiichi Manyama, <a href="/A288420/b288420.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 13 2022: (Start)

%F a(n) = A027848(n) for odd n.

%F Multiplicative with a(2^e) = (16^(e+1)-1)/15 and a(p^e) = (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1) for p > 2.

%F Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^4*zeta(5)/480 = (3/16)*zeta(4)*zeta(5) = 0.210429... . (End)

%t f[p_, e_] :=  (p^(4*e+7) - (p^3+p^2+p+1)*p^(e+1) + p^2 + p + 1)/(p^7 - (p^3+p^2+p+1)*p + p^2 + p + 1); f[2, e_] := (16^(e+1)-1)/15; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 13 2022 *)

%Y Cf. A000583, A013662, A013663, A027848, A288415.

%Y Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), A288418 (k=2), A288419 (k=3), this sequence (k=4).

%K mult,nonn

%O 1,2

%A _Seiichi Manyama_, Jun 09 2017

%E Keyword:mult added by _Andrew Howroyd_, Jul 23 2018