A358336 Multiplicative sequence with a(p^e) = ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1) for prime p and e > 0.

1, 3, 5, 12, 9, 15, 13, 40, 30, 27, 21, 60, 25, 39, 45, 120, 33, 90, 37, 108, 65, 63, 45, 200, 90, 75, 153, 156, 57, 135, 61, 336, 105, 99, 117, 360, 73, 111, 125, 360, 81, 195, 85, 252, 270, 135, 93, 600, 182, 270, 165, 300, 105, 459, 189, 520, 185, 171, 117, 540, 121, 183, 390, 896

Offset: 1

Author

Werner Schulte, Nov 09 2022

Keywords

• nonn
• easy
• mult

Crossrefs

Cf. A000010, A005361, A018804, A112526.

Cf. A001620, A002117, A013664, A244115, A157289, A306016.

Programs

Mathematica

f[p_, e_] := ((p - 1)*(1 + e*(e + 1)/2) + e)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 09 2022 *)

PARI

a(n) = { my (f=factor(n), p, e, v=1); for (k=1, #f~, p=f[k,1]; e=f[k,2]; v *= ((p-1) * (1 + e*(e+1)/2) + e) * p^(e-1)); return (v) } \\ Rémy Sigrist, Jan 18 2023

Formula

a(n) = Sum_{k=1..n} gcd(k, n) * A005361(gcd(k, n)) for n > 0.

Equals Dirichlet convolution of A000010 and n * A005361.

Dirichlet g.f.: (zeta(s-1)^2 * zeta(2*s-2) * zeta(3*s-3)) / (zeta(s) * zeta(6*s-6)).

Equals Dirichlet convolution of A018804 and A112526.

Sum_{k=1..n} a(k) ~ (zeta(3)/(2*zeta(6))) * n^2 * (log(n) + 2*gamma - 1/2 + zeta'(2)/zeta(2) + 3*zeta'(3)/zeta(3) + 6*zeta'(6)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 13 2024