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A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).

%I A373105 #11 May 26 2024 12:28:39
%S A373105 1,993,58807,1016801,9762501,58395351,282458443,1041204193,3472494301,
%T A373105 9694163493,25937263551,59795016407,137858120557,280481233899,
%U A373105 574103396307,1066193093601,2015992480593,3448186840893,6131063781703,9926520779301
%N A373105 a(n) = sigma_10(n^2)/sigma_5(n^2).
%H A373105 Amiram Eldar, <a href="/A373105/b373105.txt">Table of n, a(n) for n = 1..10000</a>
%F A373105 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) )^5.
%F A373105 a(n) = Sum_{d|n} mu(n/d) * (n/d)^5 * sigma_10(d).
%F A373105 From _Amiram Eldar_, May 25 2024: (Start)
%F A373105 Multiplicative with a(p^e) = (p^(10*e+5) + 1)/(p^5 + 1).
%F A373105 Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(s-5).
%F A373105 Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = zeta(11)/zeta(6) = 0.9834383562... . (End)
%t A373105 f[p_, e_] := (p^(10*e+5) + 1)/(p^5 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, May 25 2024 *)
%o A373105 (PARI) a(n) = sigma(n^2, 10)/sigma(n^2, 5);
%o A373105 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^5*sigma(d, 10));
%Y A373105 Cf. A057660, A084218, A084220, A372966.
%Y A373105 Cf. A371491, A371878, A373007, A373103.
%Y A373105 Cf. A013664, A013669.
%K A373105 nonn,mult
%O A373105 1,2
%A A373105 _Seiichi Manyama_, May 25 2024