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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
-
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 *)
-
PARI
a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 5));
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