A383954 a(n) = Product_{i} (phi(p_i^e_i)-1) where n = Product_{i} p_i^e_i and phi is the Euler phi function.

1, 0, 1, 1, 3, 0, 5, 3, 5, 0, 9, 1, 11, 0, 3, 7, 15, 0, 17, 3, 5, 0, 21, 3, 19, 0, 17, 5, 27, 0, 29, 15, 9, 0, 15, 5, 35, 0, 11, 9, 39, 0, 41, 9, 15, 0, 45, 7, 41, 0, 15, 11, 51, 0, 27, 15, 17, 0, 57, 3, 59, 0, 25, 31, 33, 0, 65, 15, 21, 0, 69, 15, 71, 0, 19, 17, 45, 0, 77, 21

Offset: 1

Author

Michel Marcus, Aug 19 2025

Keywords

• nonn
• mult
• easy

Comments

This is the phi- function in Sandor and Atanassof.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856.

Crossrefs

Cf. A000010 (phi), A107758 (sigma+), A057723 (sigma-), A055653 (phi+).

Programs

Mathematica

A383954[n_] := If[n == 1, 1, Times @@ (EulerPhi[Power @@@ FactorInteger[n]] - 1)];
Array[A383954, 100] (* Paolo Xausa, Aug 19 2025 *)

PARI

a(n) = my(f=factor(n)); prod(k=1, #f~, p=f[k,1]; eulerphi(f[k,1]^f[k,2])-1);

Formula

From Amiram Eldar, Aug 19 2025: (Start)

Multiplicative with a(p^e) = (p-1)*p^(e-1) - 1.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p+1) - (p+1)/p^2) = 0.39439177573628632634... . (End)