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A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.

%I A372962 #25 May 25 2024 09:02:15
%S A372962 1,29,235,925,3101,6815,16759,29597,57097,89929,160931,217375,371125,
%T A372962 486011,728735,947101,1419569,1655813,2475739,2868425,3938365,4666999,
%U A372962 6435815,6955295,9690601,10762625,13874563,15502075,20510309,21133315,28628191,30307229,37818785
%N A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^2.
%H A372962 Amiram Eldar, <a href="/A372962/b372962.txt">Table of n, a(n) for n = 1..10000</a>
%F A372962 a(n) = Sum_{d|n} mu(n/d) * (n/d)^2 * sigma_5(d).
%F A372962 From _Amiram Eldar_, May 21 2024: (Start)
%F A372962 Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1).
%F A372962 Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-2).
%F A372962 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(4) = 2*Pi^2/21 = 0.939962323... (1/A088246). (End)
%F A372962 a(n) = Sum_{d|n} phi(n/d) * (n/d)^4 * sigma_4(d^2)/sigma_2(d^2). - _Seiichi Manyama_, May 24 2024
%F A372962 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3. - _Seiichi Manyama_, May 25 2024
%t A372962 f[p_, e_] := (p^(5*e+5) - p^(5*e+2) + p^2 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372962 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^2*sigma(d, 5));
%Y A372962 Cf. A068963, A084218, A372963.
%Y A372962 Cf. A001160, A008683.
%Y A372962 Cf. A013662, A013664, A088246.
%Y A372962 Cf. A350156, A372964.
%Y A372962 Cf. A371492, A372952.
%K A372962 nonn,mult
%O A372962 1,2
%A A372962 _Seiichi Manyama_, May 18 2024