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A372927
a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.
2
1, 35, 251, 1132, 3149, 8785, 16855, 36272, 61065, 110215, 161171, 284132, 371461, 589925, 790399, 1160896, 1420145, 2137275, 2476459, 3564668, 4230605, 5640985, 6436871, 9104272, 9841225, 13001135, 14839443, 19079860, 20511989, 27663965, 28630111, 37149440, 40453921
OFFSET
1,2
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)
MATHEMATICA
f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); 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)*d^2*sigma(d, 3));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 17 2024
STATUS
approved