A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).

1, 22, 105, 394, 745, 2310, 2737, 6490, 8817, 16390, 15961, 41370, 30745, 60214, 78225, 104602, 88417, 193974, 137161, 293530, 287385, 351142, 291985, 681450, 469345, 676390, 717081, 1078378, 731641, 1720950, 953281, 1676698, 1675905, 1945174, 2039065, 3473898

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} J_2(d) * sigma(d^2), where the Jordan totient function J_2(n) = A007434(n).

From Amiram Eldar, May 26 2024: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3) * zeta(5) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^4 - 1/p^5 + 1/p^6) = 1.489916841715667671605... . (End)

Mathematica

f[p_, e_] := (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1); 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=2, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));

Crossrefs

Cf. A007434, A084218.

Cf. A062952, A373133, A373135.

Cf. A002117, A013663.

Sequence in context: A231225 A124950 A126409 * A373130 A233299 A261534

Adjacent sequences: A373129 A373130 A373131 * A373133 A373134 A373135

Keyword

nonn,mult

Author

Seiichi Manyama, May 26 2024

Status

approved