%I #18 Nov 20 2022 01:52:18

%S 1,2255,360205,8965359,195688121,812262275,11869610005,36654862063,

%T 190649623129,441276712855,2853329308061,3229367138595,21506735660905,

%U 26765970561275,70487839624805,150121132912367,548357292625505,429914900155895,2096841596815405,1754414256800439

%N a(n) = sigma_4(n^4)/sigma(n^4).

%H Amiram Eldar, <a href="/A077455/b077455.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001158(n^4)/A000203(n^4).

%F Multiplicative with a(p^e) = (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1). - _Amiram Eldar_, Sep 09 2020

%F Sum_{k=1..n} a(k) ~ c * n^13, where c = (zeta(3)*zeta(5)*zeta(9)*zeta(13)/13) * Product_{p prime} (1-1/p^2-1/p^3+1/p^5-1/p^7+1/p^8-1/p^12+2/p^13-2/p^14+2/p^15-1/p^16+2/p^17-3/p^18+1/p^19+1/p^21-1/p^22-1/p^26-1/p^27) = 0.048281563902... . - _Amiram Eldar_, Nov 20 2022

%e a(2) = sigma_4(2^4)/sigma(2^4) = 69905/31 = 2255.

%t f[p_, e_] := (p^(12*e+3) + p^(8*e+2) + p^(4*e+1) + 1)/(p^3 + p^2 + p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 20] (* _Amiram Eldar_, Sep 09 2020 *)

%o (PARI) a(n)=sumdiv(n^4,d,d^4)/sigma(n^4)

%o (PARI) a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f); \\ _Michel Marcus_, Sep 09 2020

%Y Cf. A000203, A000583, A001158, A057660, A077454, A077456.

%Y Cf. A002117, A013663, A013667, A013671.

%K nonn,easy,mult

%O 1,2

%A _Benoit Cloitre_, Nov 30 2002