A332785 Nonsquarefree numbers that are not squareful.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset

1,1

Comments

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.

This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.

From Amiram Eldar, Sep 17 2023: (Start)

Called "hybrid numbers" by Jakimczuk (2019).

These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).

Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.

Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.

The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Formula

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

Example

18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Mathematica

Select[Range[225], Max[(e = FactorInteger[#][[;; , 2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

Prog

(PARI) isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

Crossrefs

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

Sequence in context: A242416 A360248 A317616 * A177425 A182854 A348097

Adjacent sequences: A332782 A332783 A332784 * A332786 A332787 A332788

Keyword

nonn,easy

Author

Bernard Schott, Feb 24 2020

Status

approved