A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

1, 25, 217, 793, 3001, 5425, 16465, 25369, 52705, 75025, 159721, 172081, 369097, 411625, 651217, 811801, 1414945, 1317625, 2469241, 2379793, 3572905, 3993025, 6424177, 5505073, 9378001, 9227425, 12807289, 13056745, 20486761, 16280425, 28599361, 25977625, 34659457

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).

Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)

Mathematica

f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));

Crossrefs

Cf. A084218, A350156.

Cf. A068970, A084220, A372964.

Cf. A001160, A008683.

Cf. A002117, A013664, A347328.

Sequence in context: A348201 A264493 A224419 * A297864 A065939 A298131

Adjacent sequences: A372947 A372948 A372949 * A372951 A372952 A372953

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved