

A332785


Nonsquarefree numbers that are not squareful.


6



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
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 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



FORMULA

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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



