A303946 -id:A303946 - cp's OEIS frontend

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

Bernard Schott, Feb 24 2020

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)

			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.

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.

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

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

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

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

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

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

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

A132350 If n > 1 is a k-th power with k >= 2 then a(n) = 0, otherwise a(n) = 1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

N. J. A. Sloane, Nov 11 2007

nonn

			a(4) = 0 because 4 = 2^2.
a(8) = 0 because 8 = 2^3.
a(12) = 1 because 12 is not a perfect power (though it is divisible by a perfect power).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A132349, A132351, A132352, A075802, A001597.

a132350 1 = 1
a132350 n = 1 - a075802 n  -- Reinhard Zumkeller, Jun 14 2013

Table[Boole[GCD@@FactorInteger[n][[All, 2]] == 1], {n, 100}] (* Alonso del Arte, May 28 2018 *)

(a(n)=!ispower(n)); (r(nMax) = for(j=1,nMax,print1(!ispower(j)","))); r(100)

a(n) = 1 - A075802(n) for n >= 2. - R. J. Mathar, Nov 12 2007

Given the Möbius function mu(n) = A008683(n), a(n) = abs(mu(n)) unless n is in A303946. - Alonso del Arte, May 28 2018

Edited by M. F. Hasler, Jun 01 2018

A304449 Numbers that are either squarefree or a perfect power.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, May 12 2018

nonn

First differs from A072774 at a(105) = 144, A072774(105) = 145.

Apparently the 1 and the members of A062770. - R. J. Mathar, May 22 2018

Cf. A000009, A000720, A000837, A000837, A001222, A001597, A005117, A007916, A013929, A047966, A072774, A073576, A126706, A303554, A303946.

Select[Range[150],SquareFreeQ[#]||GCD@@FactorInteger[#][[All,2]]>1&]

isok(n) = issquarefree(n) || ispower(n); \\ Michel Marcus, May 13 2018

from math import isqrt
from sympy import mobius, integer_nthroot
def A304449(n):
    def f(x): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024

Union of A005117 and A001597. Complement of A303946.

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A332785 Nonsquarefree numbers that are not squareful.

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A132350 If n > 1 is a k-th power with k >= 2 then a(n) = 0, otherwise a(n) = 1.

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A304449 Numbers that are either squarefree or a perfect power.

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Author

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Crossrefs

Programs

Mathematica

PARI

Python

Formula