A346692 a(n) = phi(n) - phi(n-phi(n)), a(1) = 1.

1, 0, 1, 1, 3, 0, 5, 2, 4, 2, 9, 0, 11, 2, 2, 4, 15, 2, 17, 4, 6, 6, 21, 0, 16, 6, 12, 4, 27, -2, 29, 8, 8, 10, 14, 4, 35, 10, 16, 8, 39, 4, 41, 12, 12, 14, 45, 0, 36, 12, 14, 12, 51, 6, 32, 8, 24, 20, 57, -4, 59, 14, 18, 16, 32, -2, 65, 20, 24, 2, 69, 8, 71, 18, 16, 20, 44, 6, 77, 16

Offset

1,5

Comments

P. Erdős conjectured that a(n) > 0 on a set of asymptotic density 1, then Luca and Pomerance proved this conjecture (see link).

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Links

Table of n, a(n) for n=1..80.

Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.

Florian Luca and Carl Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions phi and sigma, Colloq. Math., Vol. 92, No. 1 (2002), pp. 111-130.

Formula

a(n) = A000010(n) - A054571(n).

If p prime, a(p) = p-2, and for k >= 2, a(p^k) = (p-1)^2 * p^(k-2).

Maple

with(numtheory):
A := seq(phi(n) - phi(n-phi(n)), n=1..100);

Mathematica

a[n_] := (phi = EulerPhi[n]) - EulerPhi[n - phi]; Array[a, 100] (* Amiram Eldar, Jul 29 2021 *)

Prog

(PARI) a(n) = if (n==1, 1, eulerphi(n) - eulerphi(n-eulerphi(n))); \\ Michel Marcus, Jul 29 2021
(Python)
from sympy import totient as phi
def a(n):
    if n == 1: return 1
    phin = phi(n)
    return phin - phi(n - phin)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 29 2021

Crossrefs

Cf. A000010, A054571.

Cf. A051487 (a(n)=0), A051488 (a(n)<0).

Sequence in context: A261628 A007431 A215447 * A159980 A343054 A098496

Adjacent sequences: A346689 A346690 A346691 * A346693 A346694 A346695

Keyword

sign

Author

Bernard Schott, Jul 29 2021

Status

approved