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A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

%I A368743 #28 Jan 29 2024 11:07:51
%S A368743 1,11,35,100,149,385,391,848,1017,1639,1451,3500,2365,4301,5215,6976,
%T A368743 5201,11187,7219,14900,13685,15961,12695,29680,19225,26015,28107,
%U A368743 39100,25229,57365,30751,56576,50785,57211,58259,101700,52021,79409,82775,126352
%N A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.
%H A368743 Amiram Eldar, <a href="/A368743/b368743.txt">Table of n, a(n) for n = 1..10000</a>
%H A368743 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%F A368743 a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^2.
%F A368743 a(n) = Sum_{d divides n} d^3 * J_2(n/d) = Sum_{d divides n} d^2 * J_3(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n).
%F A368743 Dirichlet g.f.: zeta(s-2) * zeta(s-3)/zeta(s).
%F A368743 a(n) is a multiplicative function: for prime p, a(p^k) = p^(3*k-2)*(p^2 + p + 1) - p^(2*k-2)*(p + 1).
%F A368743 Sum_{k=1..n} a(k) ~ c * n^4, where c = 15/(4*Pi^2) = 0.379954... . - _Amiram Eldar_, Jan 29 2024
%F A368743 a(n) = Sum_{d divides n} mobius(n/d) * d^2 * sigma(d). - _Peter Bala_, Jan 29 2024
%p A368743 seq( add(add(igcd(i, j, n)^3, i = 1..n), j = 1..n), n = 1..50);
%p A368743 # faster program for large n
%p A368743 with(numtheory):
%p A368743 A007434 := proc(n) add(d^2*mobius(n/d), d in divisors(n)) end proc:
%p A368743 seq( add(d^3*A007434(n/d), d in divisors(n)), n = 1..500);
%t A368743 f[p_, e_] := p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 29 2024 *)
%o A368743 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1));} \\ _Amiram Eldar_, Jan 29 2024
%o A368743 (Python)
%o A368743 from math import prod
%o A368743 from sympy import factorint
%o A368743 def A368743(n): return prod(p**(e-1<<1)*(p**e*(p*(q:=p+1)+1)-q) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jan 29 2024
%Y A368743 Cf. A007434, A059376, A069097, A343497, A343498, A360428.
%K A368743 nonn,mult,easy
%O A368743 1,2
%A A368743 _Peter Bala_, Jan 20 2024