A372931 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^4.

1, 31, 161, 736, 1249, 4991, 4801, 15616, 19521, 38719, 29281, 118496, 57121, 148831, 201089, 311296, 167041, 605151, 260641, 919264, 772961, 907711, 559681, 2514176, 1170625, 1770751, 2106081, 3533536, 1414561, 6233759, 1847041, 5963776, 4714241, 5178271, 5996449

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} mu(n/d) * d^4 * tau(d), where mu is the Moebius function A008683.

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = (e - e/p^4 + 1) * p^(4*e).

Dirichlet g.f.: zeta(s-4)^2/zeta(s).

Sum_{k=1..n} a(k) ~ (n^5/(5*zeta(5))) * (log(n) + 2*gamma - 1/5 - zeta'(5)/zeta(5)), where gamma is Euler's constant (A001620). (End)

Mathematica

f[p_, e_] := (e - e/p^4 + 1) * p^(4*e); 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^4*numdiv(d));

Crossrefs

Column k=4 of A372938.

Cf. A343498, A372926, A372929, A372937.

Cf. A000005, A008683.

Cf. A001620, A013663.

Sequence in context: A142906 A157373 A058285 * A268258 A142306 A296114

Adjacent sequences: A372928 A372929 A372930 * A372932 A372933 A372934

Keyword

nonn,mult

Author

Seiichi Manyama, May 17 2024

Status

approved