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%I #11 Jan 12 2024 22:31:13
%S 1,4,5,16,9,20,13,48,21,36,21,80,25,52,45,128,33,84,37,144,65,84,45,
%T 240,65,100,81,208,57,180,61,320,105,132,117,336,73,148,125,432,81,
%U 260,85,336,189,180,93,640,133,260,165,400,105,324,189,624,185,228,117,720,121,244,273,768
%N a(n) = Sum_{k = 1..n} gcd(4*k, n).
%F a(4*n) = 16*A018804(n); a(4*n+2) = 4*A018804(2*n+1); a(4*n+r) = A018804(4*n+r) for r = 1 and 3.
%F a(n) = Sum_{d divides n} gcd(4, d)*phi(d)*n/d, where phi(n) = A000010(n)
%F Multiplicative: a(2^k) = k*2^(k+1) for k >= 1; for odd prime p, a(p^k) = (k + 1)*p^k - k*p^(k-1).
%F Define D(n) = Sum_{d divides n} a(d). Then
%F D(4*n+r) = (4*n + r)*tau(4*n+r) for r = 1 and r = 3, where tau(n) = A000005(n), the number of divisors of n.
%F D(4*n+2) = (5/4)*(4*n + 2)*tau(4*n+2).
%F The sequence defined for n >= 1 by u(n) = (1/4)*( D(4*n) - D(n) ) begins {5, 16, 30, 44, 50, 96, 70, 112, 135, 160, 110, 264, 130, 224, 300, 272, 170, 432, 190, 440, 420, 352, ...} and appears to be multiplicative: that is, u(1)*u(n*m) = u(n)*u(m) for n and m coprime.
%F Dirichlet g.f.: (1 + 4/4^s)/(1 - 1/2^s) * zeta(s-1)^2/zeta(s).
%F Sum_{k=1..n} a(k) ~ n^2 * (5*log(n) - 5/2 + 10*gamma - 11*log(2)/3 - 30*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 12 2024
%p seq(add(gcd(4*k, n), k = 1..n), n = 1..70);
%p # alternative faster program for large n
%p with(numtheory): seq(add(gcd(4,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);
%t Table[Sum[GCD[4*k, n], {k, 1, n}], {n, 1, 100}] (* _Vaclav Kotesovec_, Jan 12 2024 *)
%Y Cf. A000010, A018804, A344372, A368736 - A368742.
%K nonn,mult,easy
%O 1,2
%A _Peter Bala_, Jan 07 2024
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