cp's OEIS Frontend

A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.

%I A372965 #24 May 22 2024 01:59:41
%S A372965 1,17,163,529,2501,2771,14407,16913,39529,42517,146411,86227,342733,
%T A372965 244919,407663,541201,1336337,671993,2345779,1323029,2348341,2488987,
%U A372965 6156503,2756819,7815001,5826461,9605467,7621303,19803869,6930271,27705631,17318417,23864993
%N A372965 a(n) = Sum_{k = 1..n} ( n/gcd(k, n) )^4.
%H A372965 Seiichi Manyama, <a href="/A372965/b372965.txt">Table of n, a(n) for n = 1..10000</a>
%F A372965 a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_5(d).
%F A372965 a(n) = Sum_{d|n} d^(5-m) * phi(d^m) for m > 0.
%F A372965 G.f.: Sum_{k>=1} k^(5-m) * phi(k^m) * x^k/(1 - x^k) for m > 0.
%F A372965 From _Amiram Eldar_, May 21 2024: (Start)
%F A372965 Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1).
%F A372965 Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-4).
%F A372965 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(2) = 2*Pi^4/315 = 0.6184704192... (1/A157292). (End)
%t A372965 f[p_, e_] := (p^(5*e+5) - p^(5*e+4) + p^4 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372965 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 5));
%o A372965 (PARI) a(n) = sumdiv(n, d, eulerphi(d^5));
%Y A372965 Cf. A057660, A068963, A068970.
%Y A372965 Cf. A001160, A008683.
%Y A372965 Cf. A372966.
%Y A372965 Cf. A013661, A013664, A157292.
%K A372965 nonn,mult
%O A372965 1,2
%A A372965 _Seiichi Manyama_, May 18 2024