A389497 Numbers whose number of powerful divisors is a square.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset

1,2

Comments

First differs from A336224 at n = 146: a(146) = 216 is not a term in A336224. Also, A336224(173) = 256 is the least term in A336224 that is not in this sequence.

Subsequence of A389499 and first differs from it at n = 97: A389499(97) = 144 is not a term in this sequence.

Numbers k for which the product of their prime factorization exponents, A005361(k), is a square.

If k and m are coprime terms, then k*m is also a term.

In particular, if k is a term, and m is a squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A389498) are the powerful (A001694) terms.

The asymptotic density of this sequence is Sum_{n>=1} f(A389498(n)) = 0.66618313..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)).

Links

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

Mathematica

q[k_] := IntegerQ[Sqrt[Times @@ FactorInteger[k][[;; , 2]]]]; Select[Range[100], q]

Prog

(PARI) isok(k) = issquare(vecprod(factor(k)[, 2]));

Crossrefs

Cf. A001694, A005361, A036436, A336224.

Subsequences: A005117, A197680, A389498.

Subsequence of A389499.

Sequence in context: A252895 A366242 A336224 * A389499 A274034 A197680

Adjacent sequences: A389494 A389495 A389496 * A389498 A389499 A389500

Keyword

nonn,easy

Author

Amiram Eldar, Oct 07 2025

Status

approved