A383954 - OEIS

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

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OFFSET

1,5

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.

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)

MATHEMATICA

A383954[n_] := If[n == 1, 1, Times @@ (EulerPhi[Power @@@ FactorInteger[n]] - 1)];

Array[A383954, 100] (* Paolo Xausa, Aug 19 2025 *)

PROG

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

CROSSREFS

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

Sequence in context: A201565 A088191 A378034 * A179179 A291503 A108500

Adjacent sequences: A383951 A383952 A383953 * A383955 A383956 A383957

KEYWORD

nonn,mult,easy

AUTHOR

Michel Marcus, Aug 19 2025

STATUS

approved