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.

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

Seiichi Manyama, Table of n, a(n) for n = 1..10000

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

Cf. A068963, A084218, A372962.

Cf. A084220.

Cf. A013663, A013665.

Sequence in context: A132388 A086945 A142365 * A251426 A232858 A373724

Adjacent sequences: A372960 A372961 A372962 * A372964 A372965 A372966

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved