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A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} n/gcd(x_1, x_2, x_3, x_4, x_5, n).

%I A371878 #26 May 26 2024 16:10:19
%S A371878 1,63,727,4031,15621,45801,117643,257983,529981,984123,1771551,
%T A371878 2930537,4826797,7411509,11356467,16510911,24137553,33388803,47045863,
%U A371878 62968251,85526461,111607713,148035867,187553641,244078121,304088211,386356147,474218933,594823293,715457421
%N A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} n/gcd(x_1, x_2, x_3, x_4, x_5, n).
%H A371878 Amiram Eldar, <a href="/A371878/b371878.txt">Table of n, a(n) for n = 1..10000</a>
%F A371878 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, x_4, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
%F A371878 a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_6(d).
%F A371878 From _Amiram Eldar_, May 25 2024: (Start)
%F A371878 Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1).
%F A371878 Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-1).
%F A371878 Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(6) = 0.9911595361106... . (End)
%t A371878 f[p_, e_] := (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, May 25 2024 *)
%o A371878 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 6));
%Y A371878 Column k=5 of A372968.
%Y A371878 Cf. A371491, A373007, A373103, A373105.
%Y A371878 Cf. A013664, A013665.
%K A371878 nonn,mult
%O A371878 1,2
%A A371878 _Seiichi Manyama_, May 25 2024