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

1,3

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).

Links

Table of n, a(n) for n=1..57.

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.

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 *)

Prog

(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)])

Crossrefs

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

Sequence in context: A256890 A110476 A379100 * A330762 A059343 A285944

Adjacent sequences: A384713 A384714 A384715 * A384717 A384718 A384719

Keyword

nonn

Author

Darío Clavijo, Jun 11 2025

Status

approved