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a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.
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%I #18 May 22 2024 01:56:49

%S 1,35,251,1132,3149,8785,16855,36272,61065,110215,161171,284132,

%T 371461,589925,790399,1160896,1420145,2137275,2476459,3564668,4230605,

%U 5640985,6436871,9104272,9841225,13001135,14839443,19079860,20511989,27663965,28630111,37149440,40453921

%N a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^5.

%H Seiichi Manyama, <a href="/A372927/b372927.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.

%F a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^2.

%F a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_3(d), where mu is the Moebius function A008683.

%F From _Amiram Eldar_, May 21 2024: (Start)

%F Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1).

%F Dirichlet g.f.: zeta(s-2)*zeta(s-5)/zeta(s).

%F Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(4)/zeta(6) = 21/(2*Pi^2) = 1.0638724... (A088246). (End)

%t f[p_, e_] := p^(2*e-2) * (p^2 * (p^(3*e+3)-1) - p^(3*e) + 1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 3));

%Y Cf. A069097, A360428, A368743, A372926.

%Y Cf. A001158, A008683.

%Y Cf. A013662, A013664, A088246.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 17 2024