cp's OEIS Frontend

A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

%I A343498 #44 Jun 24 2024 08:47:50
%S A343498 1,17,83,274,629,1411,2407,4388,6729,10693,14651,22742,28573,40919,
%T A343498 52207,70216,83537,114393,130339,172346,199781,249067,279863,364204,
%U A343498 393145,485741,545067,659518,707309,887519,923551,1123472,1216033,1420129,1514003,1843746,1874197
%N A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.
%H A343498 Seiichi Manyama, <a href="/A343498/b343498.txt">Table of n, a(n) for n = 1..10000</a>
%F A343498 a(n) = Sum_{d|n} phi(n/d) * d^4.
%F A343498 a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).
%F A343498 G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.
%F A343498 Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F A343498 Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - _Vaclav Kotesovec_, May 20 2021
%F A343498 Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - _Amiram Eldar_, Nov 22 2022
%F A343498 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - _Peter Bala_, Jan 18 2024
%t A343498 a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%t A343498 f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%o A343498 (PARI) a(n) = sum(k=1, n, gcd(k, n)^4);
%o A343498 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);
%o A343498 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 3));
%o A343498 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))
%o A343498 (Magma)
%o A343498 A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
%o A343498 [A343498(n): n in [1..50]]; // _G. C. Greubel_, Jun 24 2024
%o A343498 (SageMath)
%o A343498 def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))
%o A343498 [A343498(n) for n in range(1,51)] # _G. C. Greubel_, Jun 24 2024
%Y A343498 Column 4 of A343510.
%Y A343498 Cf. A000010, A001158 (sigma_3(n)), A018804, A054611, A069097, A332517, A342423, A342433, A343497, A343499, A343514.
%K A343498 nonn,mult
%O A343498 1,2
%A A343498 _Seiichi Manyama_, Apr 17 2021