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A373007 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) )^2.

%I A373007 #48 May 26 2024 16:10:40
%S A373007 1,125,2179,15997,78101,272375,823495,2047613,4765465,9762625,
%T A373007 19487051,34857463,62748349,102936875,170182079,262094461,410338385,
%U A373007 595683125,893871379,1249381697,1794395605,2435881375,3404824919,4461748727,6101640601,7843543625,10422071947
%N A373007 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) )^2.
%H A373007 Amiram Eldar, <a href="/A373007/b373007.txt">Table of n, a(n) for n = 1..10000</a>
%F A373007 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, x_3, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
%F A373007 a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_7(d).
%F A373007 From _Amiram Eldar_, May 25 2024: (Start)
%F A373007 Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1).
%F A373007 Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-2).
%F A373007 Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(6) = Pi^2/10 = 0.986960440108... . (End)
%t A373007 f[p_, e_] := (p^(7*e+7) - p^(7*e+2) + p^2 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 27] (* _Amiram Eldar_, May 25 2024 *)
%o A373007 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 7));
%Y A373007 Cf. A371491, A371878, A373103, A373105.
%Y A373007 Cf. A068963, A084218, A372962, A372963.
%Y A373007 Cf. A013664, A013666.
%K A373007 nonn,mult
%O A373007 1,2
%A A373007 _Seiichi Manyama_, May 25 2024