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

%I A384763 #16 Jun 15 2025 22:59:33
%S A384763 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,
%T A384763 144,18,144,28,4096,30,16,400,256,576,144,36,324,576,256,40,20736,42,
%U A384763 400,576,484,46,256,42,400,1024,576,52,324,1600,576,1296,784,58,65536
%N A384763 Product of the Euler totients of the unitary divisors of n.
%C A384763 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.
%C A384763 a(n) is upper bounded by A061537(n) (product of phi(d) over all divisors d of n).
%C A384763 The function is not multiplicative.
%C A384763 The sum of the totients over all unitary divisors d of n is A055653(n).
%F A384763 a(n) = Product_{d|n} phi(d) if gcd(n,floor(n/d)) = 1.
%F A384763 a(p) = p-1 for p prime.
%F A384763 a(p^k) = p^k-p^(k-1).
%F A384763 a(n) = phi(n)^(2^(omega(n)-1)) = A000010(n)^(A034444(n)/2). - _Amiram Eldar_, Jun 09 2025
%e A384763 For n = 6, a(6) = phi(1) * phi(2) * phi(3) * phi(6) = 1*1*2*2 = 4.
%t A384763 a[n_] := EulerPhi[n]^(2^(PrimeNu[n] - 1)); Array[a, 100] (* _Amiram Eldar_, Jun 09 2025 *)
%o A384763 (Python)
%o A384763 from sympy import totient, divisors, gcd
%o A384763 def a(n):
%o A384763    prod = 1
%o A384763    for d in divisors(n):
%o A384763       if gcd(d, n//d) == 1:
%o A384763           prod *= totient(d)
%o A384763    return prod
%o A384763 print([a(n) for n in range(1, 61)])
%o A384763 (PARI) a(n) = my(p=1); fordiv(n, d, if (gcd(d,n/d) == 1, p*=eulerphi(d))); p; \\ _Michel Marcus_, Jun 09 2025
%Y A384763 Cf. A000010, A034444, A055653, A061537, A077610.
%K A384763 nonn
%O A384763 1,3
%A A384763 _Darío Clavijo_, Jun 09 2025