login
A373130
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).
3
1, 22, 105, 414, 745, 2310, 2737, 7134, 9231, 16390, 15961, 43470, 30745, 60214, 78225, 118238, 88417, 203082, 137161, 308430, 287385, 351142, 291985, 749070, 481245, 676390, 767391, 1133118, 731641, 1720950, 953281, 1924574, 1675905, 1945174, 2039065, 3821634
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n) )^3 ).
a(n) = Sum_{d|n} J_3(d) * sigma(d), where the Jordan totient function J_3(n) = A059376(n)
From Amiram Eldar, May 26 2024: (Start)
Multiplicative with a(p^e) = (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.54488120152452251241... . (End)
MATHEMATICA
f[p_, e_] := (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + 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=3, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 26 2024
STATUS
approved