A364932 a(n) = phi(psi(n)).

1, 2, 2, 2, 2, 4, 4, 4, 4, 6, 4, 8, 6, 8, 8, 8, 6, 12, 8, 12, 16, 12, 8, 16, 8, 12, 12, 16, 8, 24, 16, 16, 16, 18, 16, 24, 18, 16, 24, 24, 12, 32, 20, 24, 24, 24, 16, 32, 24, 24, 24, 24, 18, 36, 24, 32, 32, 24, 16, 48, 30, 32, 32, 32, 24, 48, 32, 36, 32, 48, 24

Offset: 1

Author

Torlach Rush, Aug 13 2023

Keywords

• nonn
• look

Comments

Here phi is Euler's totient function and psi is the Dedekind psi function.

Values of psi(n), n > 1 are always greater than n, while values of phi(n), n > 1 are always less than n.

a(39270) = 41472 is the first term where phi(psi(n)) exceeds n.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001615, A293713, A364631.

Programs

Maple

f:= proc(n) local p; numtheory:-phi(n * mul(1+1/p, p = numtheory:-factorset(n))) end proc:
map(f, [$1..100]); # Robert Israel, Feb 13 2024

Mathematica

a[n_] := EulerPhi[n*Times @@ (1 + 1/FactorInteger[n][[;; , 1]])]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

PARI

a(n) = eulerphi(n * sumdivmult(n, d, issquarefree(d)/d)); \\ Michel Marcus, Aug 13 2023

Python

from sympy.ntheory.factor_ import totient
from sympy import isprime, primefactors, prod
def psi(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
def a(n): return totient(psi(n))

Formula

a(n) = A000010(A001615(n)).

a(2^k) = A000010(2^k), k >= 2.