A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

1, 106, 1041, 7218, 19345, 110346, 136801, 465522, 768327, 2050570, 1947121, 7513938, 5226481, 14500906, 20138145, 29822066, 25640641, 81442662, 49651921, 139632210, 142409841, 206394826, 154751521, 484608402, 302749845, 554006986, 560366223, 987429618, 616040881

Offset: 1

Author

Seiichi Manyama, May 26 2024

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A059376, A084220.

Cf. A373130, A373131.

Cf. A062952, A373132, A373135.

Cf. A013662, A013665.

Programs

Mathematica

f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

PARI

J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));

Formula

a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).

From Amiram Eldar, May 26 2024: (Start)

Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).

Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)