A368741 a(n) = Sum_{k = 1..n} gcd(5*k + 1, n).

1, 3, 5, 8, 5, 15, 13, 20, 21, 15, 21, 40, 25, 39, 25, 48, 33, 63, 37, 40, 65, 63, 45, 100, 25, 75, 81, 104, 57, 75, 61, 112, 105, 99, 65, 168, 73, 111, 125, 100, 81, 195, 85, 168, 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

Offset: 1

Author

Peter Bala, Jan 08 2024

Keywords

• nonn
• mult
• easy

Crossrefs

Cf. A000010, A011558, A018804, A368736 - A368742.

Programs

Maple

with(numtheory): seq(add(gcd(5*k+1, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(irem(d^4,5)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Mathematica

Table[Sum[GCD[5*k+1, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 12 2024 *)

Formula

a(n) = Sum_{k = 1..n} gcd(5*k + r, n) for 1 <= r <= 4.

a(5*n) = 5*a(n); a(5*n+r) = A018804(5*n+r) for 1 <= r <= 4.

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.

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).

Dirichlet g.f.: (1 - 5/5^s)/(1 - 1/5^s) * zeta(s-1)^2/zeta(s).

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