This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343509 #35 Jan 29 2024 11:02:17
%S A343509 1,129,2189,16514,78129,282381,823549,2113796,4787349,10078641,
%T A343509 19487181,36149146,62748529,106237821,171024381,270565896,410338689,
%U A343509 617568021,893871757,1290222306,1802748761,2513846349,3404825469,4627099444,6103828145,8094560241
%N A343509 a(n) = Sum_{k=1..n} gcd(k, n)^7.
%C A343509 In general, for m > 1, if a(n) = Sum_{j=1..n} gcd(j, n)^m, then Sum_{k=1..n} a(k) ~ zeta(m) * n^(m+1) / ((m+1) * zeta(m+1)). - _Vaclav Kotesovec_, May 20 2021
%H A343509 Seiichi Manyama, <a href="/A343509/b343509.txt">Table of n, a(n) for n = 1..10000</a>
%F A343509 a(n) = Sum_{d|n} phi(n/d) * d^7.
%F A343509 a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).
%F A343509 G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8.
%F A343509 Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021
%F A343509 Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - _Vaclav Kotesovec_, May 20 2021
%F A343509 Multiplicative with a(p^e) = p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1). - _Amiram Eldar_, Nov 22 2022
%F A343509 a(n) = Sum_{1 <= i_1, ..., i_7 <= n} gcd(i_1, ..., i_7, n) = Sum_{d divides n} d * J_7(n/d), where the Jordan totient function J_7(n) = A069092(n). - _Peter Bala_, Jan 29 2024
%t A343509 a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)
%t A343509 f[p_, e_] := p^(e-1)*(p^(6*e+7) - p^(6*e) - p + 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%o A343509 (PARI) a(n) = sum(k=1, n, gcd(k, n)^7);
%o A343509 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);
%o A343509 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));
%o A343509 (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+120*x^k+1191*x^(2*k)+2416*x^(3*k)+1191*x^(4*k)+120*x^(5*k)+x^(6*k))/(1-x^k)^8))
%Y A343509 Column 7 of A343510.
%Y A343509 Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.
%K A343509 nonn,mult,easy
%O A343509 1,2
%A A343509 _Seiichi Manyama_, Apr 17 2021