A375932 The largest unitary k-free divisor of n where k = A051903(n) is the maximum exponent in the prime factorization of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 11, 5, 1, 1, 3, 1, 2, 1, 13, 1, 2, 1, 7, 1, 1, 1, 15, 1, 1, 7, 1, 1, 1, 1, 17, 1, 1, 1, 9, 1, 1, 3, 19, 1, 1, 1, 5, 1, 1, 1, 21, 1

Offset: 1

Author

Amiram Eldar, Sep 03 2024

Keywords

• nonn
• easy

Comments

The product of the prime powers in the prime factorization of n that have an exponent that is smaller than the maximum exponent in this factorization.

Examples

			60 = 2^2 * 3 * 5, and the maximum exponent in the prime factorization of 60 is 2, which is the exponent of its prime factor 2. Therefore a(60) = 3 * 5 = 15.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A049606, A051903, A072774, A375931, A375933.

Programs

Mathematica

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; n / (Times @@ p[[i]])^m]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] < m, p[i]^e[i], 1)));}

Formula

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i < max_{j} e_j]), where [] is the Iverson bracket.

a(n) = n / A375931(n).

a(n) = 1 if and only if n is a power of a squarefree number (A072774).

A051903(a(n)) = A375933(n).

a(n!) = A049606(n) for n != 3.