|
| |
|
|
A343508
|
|
a(n) = Sum_{k=1..n} gcd(k, n)^6.
|
|
3
|
|
|
|
1, 65, 731, 4162, 15629, 47515, 117655, 266372, 532905, 1015885, 1771571, 3042422, 4826821, 7647575, 11424799, 17047816, 24137585, 34638825, 47045899, 65047898, 86005805, 115152115, 148035911, 194717932, 244203145, 313743365, 388487763, 489680110, 594823349
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
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
|
Column 6 of A343510.
Cf. A000010, A001160 (sigma_5(n)), A343520.
Sequence in context: A017675 A013954 A294301 * A116277 A220389 A196634
Adjacent sequences: A343505 A343506 A343507 * A343509 A343510 A343511
|
|
|
KEYWORD
|
nonn,mult
|
|
|
AUTHOR
|
Seiichi Manyama, Apr 17 2021
|
|
|
STATUS
|
approved
|
| |
|
|
