OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|n} phi(n/d) * d^6.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_5(d).
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.
Dirichlet g.f.: zeta(s-1) * zeta(s-6) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^6 * n^7 / (6615*zeta(7)). - Vaclav Kotesovec, May 20 2021
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
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
MATHEMATICA
a[n_] := Sum[GCD[k, n]^6, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
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 *)
PROG
(PARI) a(n) = sum(k=1, n, gcd(k, n)^6);
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*d^6);
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 5));
(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))
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Seiichi Manyama, Apr 17 2021
STATUS
approved