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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).
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%I #24 Nov 29 2022 02:41:30

%S 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 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 1,1,1,2,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1

%N 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 The unitary version of A072911.

%C 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 Amiram Eldar, <a href="/A321167/b321167.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicusor Minculete and László Tóth, <a href="http://ac.inf.elte.hu/Vol_035_2011/elte_annales_35_jav_vagott.pdf#page=205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

%F 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 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 (PARI) uphi(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1)};

%o a(n) = {my(f=factor(n)); prod(i=1, #f~, uphi(f[i,2]))}; \\ _Amiram Eldar_, Nov 29 2022

%Y Cf. A047994, A072911, A358658.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Jan 10 2019