A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).

1, 50, 339, 1786, 3845, 16950, 19495, 58682, 85281, 192250, 176891, 605454, 401869, 974750, 1303455, 1890106, 1507985, 4264050, 2612899, 6867170, 6608805, 8844550, 6727799, 19893198, 12109345, 20093450, 20802003, 34818070, 21241949, 65172750, 29581471, 60581690

Offset

1,2

Links

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

Formula

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

a(n) = Sum_{d|n} J_3(d) * sigma(d^2), where the Jordan totient function J_3(n) = A059376(n).

From Amiram Eldar, May 26 2024: (Start)

Multiplicative with a(p^e) = (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4) = 1.67666099579383196077... . (End)

Mathematica

f[p_, e_] := (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2024 *)

Prog

(PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
a(n, k=3, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

Crossrefs

Cf. A059376, A372962.

Cf. A373130, A373133.

Cf. A013664.

Sequence in context: A261803 A184564 A184556 * A334697 A280548 A293608

Adjacent sequences: A373128 A373129 A373130 * A373132 A373133 A373134

Keyword

nonn,mult

Author

Seiichi Manyama, May 26 2024

Status

approved