A343509 - OEIS

%I #35 Jan 29 2024 11:02:17

%S 1,129,2189,16514,78129,282381,823549,2113796,4787349,10078641,

%T 19487181,36149146,62748529,106237821,171024381,270565896,410338689,

%U 617568021,893871757,1290222306,1802748761,2513846349,3404825469,4627099444,6103828145,8094560241

%N a(n) = Sum_{k=1..n} gcd(k, n)^7.

%C 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 Seiichi Manyama, <a href="/A343509/b343509.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(n/d) * d^7.

%F a(n) = Sum_{d|n} mu(n/d) * d * sigma_6(d).

%F 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 Dirichlet g.f.: zeta(s-1) * zeta(s-7) / zeta(s). - _Ilya Gutkovskiy_, Apr 18 2021

%F Sum_{k=1..n} a(k) ~ 4725*zeta(7)*n^8 / (4*Pi^8). - _Vaclav Kotesovec_, May 20 2021

%F 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 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 a[n_] := Sum[GCD[k, n]^7, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, Apr 18 2021 *)

%t 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 (PARI) a(n) = sum(k=1, n, gcd(k, n)^7);

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^7);

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 6));

%o (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 Column 7 of A343510.

%Y Cf. A000010, A013954 (sigma_6(n)), A069092, A343521.

%K nonn,mult,easy

%O 1,2

%A _Seiichi Manyama_, Apr 17 2021