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).

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, 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, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 10 2019

Keywords

• nonn
• easy
• mult

Comments

The unitary version of A072911.

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.

Crossrefs

Cf. A047994, A072911, A358658.

Programs

Mathematica

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]

PARI

uphi(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1)};
a(n) = {my(f=factor(n)); prod(i=1, #f~, uphi(f[i,2]))}; \\ Amiram Eldar, Nov 29 2022

Formula

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