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

1,6

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))).

Links

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

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.

Example

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).

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A161097 A105240 A327499 * A327857 A363850 A227481

Adjacent sequences: A353690 A353691 A353692 * A353694 A353695 A353696

Keyword

nonn

Author

Amiram Eldar, May 04 2022

Status

approved