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%I #11 Jan 12 2024 22:31:41
%S 1,3,5,8,5,15,13,20,21,15,21,40,25,39,25,48,33,63,37,40,65,63,45,100,
%T 25,75,81,104,57,75,61,112,105,99,65,168,73,111,125,100,81,195,85,168,
%U 105,135,93,240,133,75,165,200,105,243,105,260,185,171,117,200,121,183,273,256,125,315,133,264,225,195
%N a(n) = Sum_{k = 1..n} gcd(5*k + 1, n).
%F a(n) = Sum_{k = 1..n} gcd(5*k + r, n) for 1 <= r <= 4.
%F a(5*n) = 5*a(n); a(5*n+r) = A018804(5*n+r) for 1 <= r <= 4.
%F a(n) = Sum_{d divides n} X(d)*phi(d)*n/d, where phi(n) = A000010(n) and X(n) = A011558(n) is the principal Dirichlet character of the reduced residue system mod 5.
%F Multiplicative: a(5^k) = 5^k and for prime p not equal to 5, a(p^k) = (k + 1)*p^k - k*p^(k-1).
%F Dirichlet g.f.: (1 - 5/5^s)/(1 - 1/5^s) * zeta(s-1)^2/zeta(s).
%F Sum_{k=1..n} a(k) ~ 5*n^2 * (log(n)/2 - 1/4 + gamma + 5*log(5)/48 - 3*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 12 2024
%p with(numtheory): seq(add(gcd(5*k+1, n), k = 1..n), n = 1..70);
%p # alternative faster program for large n
%p with(numtheory): seq(add(irem(d^4,5)*phi(d)*n/d, d in divisors(n)), n = 1..70);
%t Table[Sum[GCD[5*k+1, n], {k, 1, n}], {n, 1, 100}] (* _Vaclav Kotesovec_, Jan 12 2024 *)
%Y Cf. A000010, A011558, A018804, A368736 - A368742.
%K nonn,mult,easy
%O 1,2
%A _Peter Bala_, Jan 08 2024
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