A353693 a(n) is the least multiplier k such that the exponents in the prime factorization of k*n are mutually distinct (A130091).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 3, 2, 5, 2, 1, 2, 3, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 2, 1, 6, 1, 2, 1, 1, 5, 12, 1, 1, 3, 20, 1, 1, 1, 2, 1, 1, 7, 12, 1, 1, 1, 2, 1, 6, 5, 2

Offset: 1

Author

Amiram Eldar, May 04 2022

Keywords

• nonn

Comments

First differs from A327499 at n = 30.

If n = Product_{i=1..k} p_i is squarefree (A005117), and p_1 < p_2 < ... < p_k are its k ordered prime divisors, then a(n) = Product_{i} p_i^(k-i).

If n is powerful (A001694) then a(n) = a(n/rad(n)), where rad(n) is the squarefree kernel of n (A007947). In general, if k = A051904(n) is the minimal exponent in the prime factorization of n, then a(n) = a(n/(rad(n)^(k-1))).

Examples

			a(2) = 1 since 2 = 2^1 has only one exponent (1) in its prime factorization.
a(6) = 2 since 6 = 2*3 has two equal exponents (1) in its prime factorization, and 2*6 = 12 = 2^2*3 has two distinct exponents (1 and 2).
		

Links

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

Crossrefs

Cf. A001694, A005117, A007947, A130091, A130092, A327498, A327499, A353694.

Programs

Mathematica

a[n_] := Module[{k = 1}, While[!UnsameQ @@ FactorInteger[k*n][[;; , 2]], k++]; k]; Array[a, 100]

PARI

a(n) = my(k=1, f=factor(n)[,2]); while(#Set(f) != #f, k++; f=factor(k*n)[,2]); k; \\ Michel Marcus, May 05 2022

Formula

a(n) = 1 if and only if n is in A130091.

a(A130092(n)) > 1.

rad(a(n)) | rad(n).

a(n) = A353694(n)/n.