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A343508 a(n) = Sum_{k=1..n} gcd(k, n)^6.

%I A343508 #31 Jan 29 2024 11:02:20
%S A343508 1,65,731,4162,15629,47515,117655,266372,532905,1015885,1771571,
%T A343508 3042422,4826821,7647575,11424799,17047816,24137585,34638825,47045899,
%U A343508 65047898,86005805,115152115,148035911,194717932,244203145,313743365,388487763,489680110,594823349
%N A343508 a(n) = Sum_{k=1..n} gcd(k, n)^6.
%H A343508 Seiichi Manyama, <a href="/A343508/b343508.txt">Table of n, a(n) for n = 1..10000</a>
%F A343508 a(n) = Sum_{d|n} phi(n/d) * d^6.
%F A343508 a(n) = Sum_{d|n} mu(n/d) * d * sigma_5(d).
%F A343508 G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 57*x^k + 302*x^(2*k) + 302*x^(3*k) + 57*x^(4*k) + x^(5*k))/(1 - x^k)^7.
%F A343508 Dirichlet g.f.: zeta(s-1) * zeta(s-6) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F A343508 Sum_{k=1..n} a(k) ~ Pi^6 * n^7 / (6615*zeta(7)). - _Vaclav Kotesovec_, May 20 2021
%F A343508 Multiplicative with a(p^e) = p^(e-1)*(p^(5*e+6) - p^(5*e) - p + 1)/(p^5-1). - _Amiram Eldar_, Nov 22 2022
%F A343508 a(n) = Sum_{1 <= i_1, ..., i_6 <= n} gcd(i_1, ..., i_6, n) = Sum_{d divides n} d * J_6(n/d), where the Jordan totient function J_6(n) = A069091(n). - _Peter Bala_, Jan 29 2024
%t A343508 a[n_] := Sum[GCD[k, n]^6, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%t A343508 f[p_, e_] := p^(e-1)*(p^(5*e+6) - p^(5*e) - p + 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%o A343508 (PARI) a(n) = sum(k=1, n, gcd(k, n)^6);
%o A343508 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^6);
%o A343508 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 5));
%o A343508 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+57*x^k+302*x^(2*k)+302*x^(3*k)+57*x^(4*k)+x^(5*k))/(1-x^k)^7))
%Y A343508 Column 6 of A343510.
%Y A343508 Cf. A000010, A001160 (sigma_5(n)), A069091, A343520.
%K A343508 nonn,mult,easy
%O A343508 1,2
%A A343508 _Seiichi Manyama_, Apr 17 2021