A069193 a(n) = Sum_{d|n} d*phi(n)/phi(d).

1, 3, 5, 10, 9, 15, 13, 28, 24, 27, 21, 50, 25, 39, 45, 72, 33, 72, 37, 90, 65, 63, 45, 140, 70, 75, 99, 130, 57, 135, 61, 176, 105, 99, 117, 240, 73, 111, 125, 252, 81, 195, 85, 210, 216, 135, 93, 360, 140, 210, 165, 250, 105, 297, 189, 364, 185, 171, 117, 450

Offset: 1

Author

Vladeta Jovovic, Apr 10 2002

Keywords

• mult
• easy
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010.

Programs

Magma

[&+[d*EulerPhi(n) div EulerPhi(d):d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Sep 15 2019

Maple

with(numtheory):for n from 1 to 200 do di := sort(convert(divisors(n),list)): a[n] := sum(di[i]*phi(n)/phi(di[i]),i=1..nops(di)):od:seq(a[k],k=1..200);

Mathematica

f[p_, e_] := ((e+1)*p - 1) * p^(e-1) ; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2019 *)

PARI

a(n) = sumdiv(n, d, d*eulerphi(n)/eulerphi(d)); \\ Michel Marcus, Sep 15 2019

Formula

Multiplicative with a(p^e) = ((e+1)*p - 1) * p^(e-1). - Amiram Eldar, Sep 15 2019

a(n) = Sum_{k=1..n, gcd(n,k) = 1} sigma(gcd(n,k-1)). - Ilya Gutkovskiy, Sep 24 2021

Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1)). - Amiram Eldar, Sep 21 2023

Extensions

More terms from Sascha Kurz, Feb 02 2003