A384763 Product of the Euler totients of the unitary divisors of n.

1, 1, 2, 2, 4, 4, 6, 4, 6, 16, 10, 16, 12, 36, 64, 8, 16, 36, 18, 64, 144, 100, 22, 64, 20, 144, 18, 144, 28, 4096, 30, 16, 400, 256, 576, 144, 36, 324, 576, 256, 40, 20736, 42, 400, 576, 484, 46, 256, 42, 400, 1024, 576, 52, 324, 1600, 576, 1296, 784, 58, 65536

Offset: 1

Author

Darío Clavijo, Jun 09 2025

Keywords

• nonn

Comments

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

a(n) is upper bounded by A061537(n) (product of phi(d) over all divisors d of n).

The function is not multiplicative.

The sum of the totients over all unitary divisors d of n is A055653(n).

Examples

			For n = 6, a(6) = phi(1) * phi(2) * phi(3) * phi(6) = 1*1*2*2 = 4.
		

Crossrefs

Cf. A000010, A034444, A055653, A061537, A077610.

Programs

Mathematica

a[n_] := EulerPhi[n]^(2^(PrimeNu[n] - 1)); Array[a, 100] (* Amiram Eldar, Jun 09 2025 *)

PARI

a(n) = my(p=1); fordiv(n, d, if (gcd(d,n/d) == 1, p*=eulerphi(d))); p; \\ Michel Marcus, Jun 09 2025

Python

from sympy import totient, divisors, gcd
def a(n):
   prod = 1
   for d in divisors(n):
      if gcd(d, n//d) == 1:
          prod *= totient(d)
   return prod
print([a(n) for n in range(1, 61)])

Formula

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

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

a(p^k) = p^k-p^(k-1).

a(n) = phi(n)^(2^(omega(n)-1)) = A000010(n)^(A034444(n)/2). - Amiram Eldar, Jun 09 2025