A372961 - OEIS

%I #20 May 25 2024 09:02:24

%S 1,31,241,991,3121,7471,16801,31711,58561,96751,161041,238831,371281,

%T 520831,752161,1014751,1419841,1815391,2476081,3092911,4049041,

%U 4992271,6436321,7642351,9753121,11509711,14230321,16649791,20511121,23316991,28629121,32472031,38810881

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

%H Amiram Eldar, <a href="/A372961/b372961.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Amiram Eldar_, May 21 2024: (Start)

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

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

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

%F 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

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 5));

%Y Column k=4 of A372968.

%Y Cf. A001160, A008683.

%Y Cf. A013663, A013664.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 18 2024