A372966 - OEIS

%I #22 May 20 2024 02:28:11

%S 1,241,6481,61681,390001,1561921,5762401,15790321,42521761,93990241,

%T 214344241,399754561,815702161,1388738641,2527596481,4042322161,

%U 6975673921,10247744401,16983432721,24055651681,37346120881,51656962081,78310705441,102337070401

%N a(n) = sigma_8(n^2)/sigma_4(n^2).

%C Apparently, a(n) == 1 (mod 240). - _Hugo Pfoertner_, May 20 2024

%H Amiram Eldar, <a href="/A372966/b372966.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^4.

%F a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_8(d).

%F From _Amiram Eldar_, May 20 2024: (Start)

%F Multiplicative with a(p^e) = (p^(8*e + 4) + 1)/(p^4 + 1).

%F Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-4).

%F Sum_{k=1..n} a(k) ~ (zeta(9)/(9*zeta(5))) * n^9. (End)

%t f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* _Amiram Eldar_, May 20 2024 *)

%o (PARI) a(n) = sigma(n^2, 8)/sigma(n^2, 4);

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 8));

%Y Cf. A057660, A084218, A084220, A108223.

%Y Cf. A008683, A013956.

%Y Cf. A372965.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 18 2024