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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).

%I A372961 #20 May 25 2024 09:02:24
%S A372961 1,31,241,991,3121,7471,16801,31711,58561,96751,161041,238831,371281,
%T A372961 520831,752161,1014751,1419841,1815391,2476081,3092911,4049041,
%U A372961 4992271,6436321,7642351,9753121,11509711,14230321,16649791,20511121,23316991,28629121,32472031,38810881
%N 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).
%H A372961 Amiram Eldar, <a href="/A372961/b372961.txt">Table of n, a(n) for n = 1..10000</a>
%F A372961 a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_5(d).
%F A372961 From _Amiram Eldar_, May 21 2024: (Start)
%F A372961 Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+1) + p - 1)/(p^5-1).
%F A372961 Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-1).
%F A372961 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(5) = 0.981112769... . (End)
%F A372961 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 A372961 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 A372961 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 5));
%Y A372961 Column k=4 of A372968.
%Y A372961 Cf. A001160, A008683.
%Y A372961 Cf. A013663, A013664.
%K A372961 nonn,mult
%O A372961 1,2
%A A372961 _Seiichi Manyama_, May 18 2024