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

1,12

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.

Links

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

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.

Example

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.

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]

Prog

(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))); }

Crossrefs

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

Sequence in context: A359196 A348969 A030380 * A066636 A137578 A200146

Adjacent sequences: A375929 A375930 A375931 * A375933 A375934 A375935

Keyword

nonn,easy

Author

Amiram Eldar, Sep 03 2024

Status

approved