A384716 The totient of the product of unitary divisors of n.

1, 1, 2, 2, 4, 12, 6, 4, 6, 40, 10, 48, 12, 84, 120, 8, 16, 108, 18, 160, 252, 220, 22, 192, 20, 312, 18, 336, 28, 216000, 30, 16, 660, 544, 840, 432, 36, 684, 936, 640, 40, 889056, 42, 880, 1080, 1012, 46, 768, 42, 1000, 1632, 1248, 52, 972, 2200, 1344, 2052

Offset: 1

Author

Darío Clavijo, Jun 11 2025

Keywords

• nonn

Comments

a(n) is the totient of the product over all unitary divisors d of n; i.e., those divisors satisfying gcd(d, n/d) = 1.

Growth rate of a(n) is ~ n^O(log log n).

Also, a(n) is lower bounded by A000010(n).

Crossrefs

Cf. A000010, A000225, A006093, A061537, A384763.

Programs

Mathematica

f[p_, e_, m_] := (p-1)*p^(e*m-1); a[n_] := Module[{fct = FactorInteger[n]}, Times @@ (f[#1, #2, 2^(Length[fct]-1)] & @@@ fct)]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

Python

from sympy import factorint
def a(n):
    if n == 1: return 1
    factors = factorint(n)
    phi, w = 1, len(factors)
    for p, e in factors.items():
        phi *= (p - 1) * p**(e - 1)
    return n**((1 << (w-1)) - 1) * phi
print([a(n) for n in range(1, 58)])

Formula

a(n) = phi(Product_{d|n} d if gcd(d, n/d) = 1).

a(n) = n^(2^(omega(n)-1)-1) * phi(n).

a(n) = A000010(A061537(n)).

a(p) = p-1 for p prime.