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A372950
a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.
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
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
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 18 2024
STATUS
approved