cp's OEIS Frontend

A368740 a(n) = Sum_{k = 1..n} gcd(5*k, n).

%I A368740 #13 Jan 12 2024 22:31:28
%S A368740 1,3,5,8,25,15,13,20,21,75,21,40,25,39,125,48,33,63,37,200,65,63,45,
%T A368740 100,225,75,81,104,57,375,61,112,105,99,325,168,73,111,125,500,81,195,
%U A368740 85,168,525,135,93,240,133,675,165,200,105,243,525,260,185,171,117,1000,121,183,273,256,625
%N A368740 a(n) = Sum_{k = 1..n} gcd(5*k, n).
%C A368740 a(n) equals the number of solutions to the congruence 5*x*y == 0 (mod n) for 1 <= x, y <= n.
%F A368740 a(5*n) = 25*A018804(n); a(5*n+r) = A018804(5*n+r) for 1 <= r <= 4.
%F A368740 a(n) = Sum_{d divides n} gcd(5, d)*phi(d)*n/d, where phi(n) = A000010(n).
%F A368740 Multiplicative: a(5^k) = (4*k + 1)*5^k and for prime p not equal to 5, a(p^k) = (k + 1)*p^k - k*p^(k-1).
%F A368740 Define D(n) = Sum_{d divides n} a(d). Then
%F A368740 D(5*n+r) = (5*n + r)*tau(5*n+r) for 1 <= r <= 4, where tau(n) = A000005(n), the number of divisors of n.
%F A368740 The sequence {(1/25)*( D(5*n) - D(n) ) : n >= 1} begins {1, 4, 6, 12, 9, 24, 14, 32, 27, 36, 22, 72, 26, 56, 54, 80, 34, 108, 38, 108, 84, 88, ...} and appears to be multiplicative.
%F A368740 Dirichlet g.f.: (1 + 15/5^s)/(1 - 1/5^s) * zeta(s-1)^2/zeta(s).
%F A368740 Sum_{k=1..n} a(k) ~ n^2 * (5*log(n) - 5/2 + 10*gamma - 25*log(5)/12 - 30*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 12 2024
%e A368740 a(10) = 75: each of the 100 pairs (x, y), 1 <= x, y <= 10, is a solution to the congruence 5*x*y == 0 (mod 10) except for the 25 pairs (x, y) with both x and y odd.
%p A368740 seq(add(gcd(5*k, n), k = 1..n), n = 1..70);
%p A368740 # alternative faster program for large n
%p A368740 with(numtheory): seq(add(gcd(5,d)*phi(d)*n/d, d in divisors(n)), n = 1..70);
%t A368740 Table[Sum[GCD[5*k, n], {k, 1, n}], {n, 1, 100}] (* _Vaclav Kotesovec_, Jan 12 2024 *)
%Y A368740 Cf. A000010, A018804, A344372, A368736 - A368742.
%K A368740 nonn,mult,easy
%O A368740 1,2
%A A368740 _Peter Bala_, Jan 07 2024