A384763 - OEIS

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

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OFFSET

1,3

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).

LINKS

Table of n, a(n) for n=1..60.

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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)])

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

CROSSREFS

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

Sequence in context: A171462 A390608 A075857 * A023817 A300754 A145392

Adjacent sequences: A384760 A384761 A384762 * A384764 A384765 A384766

KEYWORD

nonn

AUTHOR

Darío Clavijo, Jun 09 2025

STATUS

approved