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A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.

%I A372963 #23 May 25 2024 09:02:40
%S A372963 1,61,721,3901,15601,43981,117601,249661,525601,951661,1771441,
%T A372963 2812621,4826641,7173661,11248321,15978301,24137281,32061661,47045521,
%U A372963 60859501,84790321,108057901,148035361,180005581,243765601,294425101,383163121,458761501,594822481,686147581
%N A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^2.
%H A372963 Seiichi Manyama, <a href="/A372963/b372963.txt">Table of n, a(n) for n = 1..10000</a>
%F A372963 a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_6(d).
%F A372963 From _Amiram Eldar_, May 21 2024: (Start)
%F A372963 Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1).
%F A372963 Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-2).
%F A372963 Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(5) = 0.972439277... . (End)
%F A372963 a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_6(d^2)/sigma_3(d^2). - _Seiichi Manyama_, May 24 2024
%F A372963 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - _Seiichi Manyama_, May 25 2024
%t A372963 f[p_, e_] := (p^(6*e+6) - p^(6*e+2) + p^2 - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372963 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 6));
%Y A372963 Cf. A068963, A084218, A372962.
%Y A372963 Cf. A084220.
%Y A372963 Cf. A013663, A013665.
%K A372963 nonn,mult
%O A372963 1,2
%A A372963 _Seiichi Manyama_, May 18 2024