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

%I A384716 #28 Aug 11 2025 10:24:19
%S A384716 1,1,2,2,4,12,6,4,6,40,10,48,12,84,120,8,16,108,18,160,252,220,22,192,
%T A384716 20,312,18,336,28,216000,30,16,660,544,840,432,36,684,936,640,40,
%U A384716 889056,42,880,1080,1012,46,768,42,1000,1632,1248,52,972,2200,1344,2052
%N A384716 The totient of the product of unitary divisors of n.
%C A384716 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.
%C A384716 Growth rate of a(n) is ~ n^O(log log n).
%C A384716 Also, a(n) is lower bounded by A000010(n).
%F A384716 a(n) = phi(Product_{d|n} d if gcd(d, n/d) = 1).
%F A384716 a(n) = n^(2^(omega(n)-1)-1) * phi(n).
%F A384716 a(n) = A000010(A061537(n)).
%F A384716 a(p) = p-1 for p prime.
%t A384716 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 *)
%o A384716 (Python)
%o A384716 from sympy import factorint
%o A384716 def a(n):
%o A384716     if n == 1: return 1
%o A384716     factors = factorint(n)
%o A384716     phi, w = 1, len(factors)
%o A384716     for p, e in factors.items():
%o A384716         phi *= (p - 1) * p**(e - 1)
%o A384716     return n**((1 << (w-1)) - 1) * phi
%o A384716 print([a(n) for n in range(1, 58)])
%Y A384716 Cf. A000010, A000225, A006093, A061537, A384763.
%K A384716 nonn
%O A384716 1,3
%A A384716 _Darío Clavijo_, Jun 11 2025