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A372963
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.
5
1, 61, 721, 3901, 15601, 43981, 117601, 249661, 525601, 951661, 1771441, 2812621, 4826641, 7173661, 11248321, 15978301, 24137281, 32061661, 47045521, 60859501, 84790321, 108057901, 148035361, 180005581, 243765601, 294425101, 383163121, 458761501, 594822481, 686147581
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_6(d).
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1).
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-2).
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(5) = 0.972439277... . (End)
a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024
MATHEMATICA
f[p_, e_] := (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1); 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)^2*sigma(d, 6));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 18 2024
STATUS
approved