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%I #43 Jun 24 2024 08:47:53
%S 1,33,245,1058,3129,8085,16813,33860,59541,103257,161061,259210,
%T 371305,554829,766605,1083528,1419873,1964853,2476117,3310482,4119185,
%U 5315013,6436365,8295700,9778145,12253065,14468481,17788154,20511177,25297965,28629181,34672912,39459945,46855809
%N a(n) = Sum_{k=1..n} gcd(k, n)^5.
%H Seiichi Manyama, <a href="/A343499/b343499.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} phi(n/d) * d^5.
%F a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
%F G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
%F Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - _Vaclav Kotesovec_, May 20 2021
%F Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - _Amiram Eldar_, Nov 22 2022
%F a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - _Peter Bala_, Jan 29 2024
%t a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%t f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%o (PARI) a(n) = sum(k=1, n, gcd(k, n)^5);
%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);
%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))
%o (Magma)
%o A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
%o [A343499(n): n in [1..50]]; // _G. C. Greubel_, Jun 24 2024
%o (SageMath)
%o def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
%o [A343499(n) for n in range(1,51)] # _G. C. Greubel_, Jun 24 2024
%Y Column 5 of A343510.
%Y Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.
%K nonn,mult,easy
%O 1,2
%A _Seiichi Manyama_, Apr 17 2021
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