A372961 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} n/gcd(x_1, x_2, x_3, x_4, n).

1, 31, 241, 991, 3121, 7471, 16801, 31711, 58561, 96751, 161041, 238831, 371281, 520831, 752161, 1014751, 1419841, 1815391, 2476081, 3092911, 4049041, 4992271, 6436321, 7642351, 9753121, 11509711, 14230321, 16649791, 20511121, 23316991, 28629121, 32472031, 38810881

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_5(d).

From _Amiram Eldar_, May 21 2024: (Start)

Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+1) + p - 1)/(p^5-1).

Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-1).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(5) = 0.981112769... . (End)

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - _Seiichi Manyama_, May 25 2024

Mathematica

f[p_, e_] := (p^(5*e+5) - p^(5*e+1) + p - 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*sigma(d, 5));

Crossrefs

Column k=4 of A372968.

Cf. A001160, A008683.

Cf. A013663, A013664.

Sequence in context: A059899 A140846 A082544 * A173832 A272162 A338893

Adjacent sequences: A372957 A372958 A372959 * A372962 A372963 A372964

Keyword

nonn,mult

Author

_Seiichi Manyama_, May 18 2024

Status

approved