A077455 a(n) = sigma_4(n^4)/sigma(n^4).

1, 2255, 360205, 8965359, 195688121, 812262275, 11869610005, 36654862063, 190649623129, 441276712855, 2853329308061, 3229367138595, 21506735660905, 26765970561275, 70487839624805, 150121132912367, 548357292625505, 429914900155895, 2096841596815405, 1754414256800439

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

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

Example

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

Mathematica

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 *)

Prog

(PARI) a(n)=sumdiv(n^4, d, d^4)/sigma(n^4)
(PARI) a(n) = my(f=factor(n^4)); sigma(f, 4)/sigma(f); \\ Michel Marcus, Sep 09 2020

Crossrefs

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

Cf. A002117, A013663, A013667, A013671.

Sequence in context: A287701 A123297 A251944 * A068754 A222951 A035873

Adjacent sequences: A077452 A077453 A077454 * A077456 A077457 A077458

Keyword

nonn,easy,mult

Author

Benoit Cloitre, Nov 30 2002

Status

approved