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A321167 The e-unitary Euler function: a(1) = 1, a(n) = Product uphi(e(i)) for n = Product p(i)^e(i), where uphi is the unitary totient function (A047994).

%I A321167 #27 May 30 2025 03:50:26
%S A321167 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,4,1,1,
%T A321167 1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A321167 1,1,1,2,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1
%N A321167 The e-unitary Euler function: a(1) = 1, a(n) = Product uphi(e(i)) for n = Product p(i)^e(i), where uphi is the unitary totient function (A047994).
%C A321167 The unitary version of A072911.
%C A321167 For n = Product p(i)^e(i) > 1, a(n) is the number of divisors d of n such that d and n are exponentially unitary coprime, i.e., d = Product p(i)^f(i) where 1 <= f(i) <= e(i) and uGCD(f(i), e(i)) = 1 for any i, where uGCD(m, n) is the largest divisor of m that is a unitary divisor of n.
%H A321167 Amiram Eldar, <a href="/A321167/b321167.txt">Table of n, a(n) for n = 1..10000</a>
%H A321167 Nicuşor Minculete and László Tóth, <a href="https://doi.org/10.71352/ac.35.205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.
%F A321167 Sum_{k=1..n} a(k) ~ c_1 * n + c_2 * n^(1/3) + O(n^(1/4 + eps)), where c_1 = A358658 and c_2 is a constant (see Minculete and Tóth, 2011). - _Amiram Eldar_, Nov 29 2022
%t A321167 f[p_, e_] := p^e-1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Array[euphi, 100]
%o A321167 (PARI) uphi(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1)};
%o A321167 a(n) = {my(f=factor(n)); prod(i=1, #f~, uphi(f[i,2]))}; \\ _Amiram Eldar_, Nov 29 2022
%Y A321167 Cf. A047994, A072911, A358658.
%K A321167 nonn,easy,mult
%O A321167 1,8
%A A321167 _Amiram Eldar_, Jan 10 2019