A210490 -id:A210490 - cp's OEIS frontend

A340682 The closure under squaring of the nonunit squarefree numbers.

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset: 1

Antti Karttunen and Peter Munn, Feb 07 2021

Numbers of the form s^(2^e), where s is a nonunit squarefree number, and e >= 0.
The categorization provided by this sequence and its complement, A340681, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.
All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has only one term, that is numbers m such that A331591(m) = 1.
Presence in the sequence is determined by prime signature. The set of represented signatures starts: {{1}, {2}, {1,1}, {1,1,1}, {4}, {2,2}, {1,1,1,1}, {1,1,1,1,1}, {2,2,2}, {1,1,1,1,1,1}, {1,1,1,1,1,1,1}, {8}, {4,4}, {2,2,2,2}, {1,1,1,1,1,1,1,1}, ...}. Representing each signature in the set by the least number with that signature, we get the set A133492.
Positions of terms > 1 in A340675.

			12 = 3 * 4 = 3^1 * 2^2 = 3^(2^0) * 2^(2^1). This is the (unique) factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 12 is not in the sequence.
The equivalent factorization for 36 is 36 = 6^2 = 6^(2^1). As this factorization has only 1 term, 36 is in the sequence.

Index entries for sequences related to prime signature

Cf. A340675.
Cf. A340681 (complement, apart from 1 which is in neither).
Subsequence of A072774, A210490.
Positions of ones in A331591.
Union of A005117 \ {1} and A340674.
Cf. subsequences: A050376, A133492.

Select[Range[2, 120], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* Amiram Eldar, Feb 13 2021 *)

isA340682(n) = if(!issquare(n), issquarefree(n), (n>1)&&isA340682(sqrtint(n)));

from math import isqrt
from sympy import mobius, integer_nthroot
def A340682(n):
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x-sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

Original entry on oeis.org

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180

Offset: 1

Michael Minic (Rassilon6(AT)aol.com)

At least one exponent in the canonical prime factorization (cf. A124010) of n is odd, and at least one exponent is greater than 1. - Reinhard Zumkeller, Jan 24 2013

Compare this sequence, as a set, with A177712, numbers that have an odd factor, but are not odd. The self-inverse function defined by A225546, maps the members of either one of these sets 1:1 onto the other set. - Peter Munn, Jul 31 2020

			63 is included because 63 = 3^2 * 7.
64 is not included because it is a perfect square (8^2).
65 is not included because it is squarefree (5 * 13).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Number.
Eric Weisstein's World of Mathematics, Squarefree.
Wikipedia, Square number.
Wikipedia, Squarefree integer.

Cf. A210490 (complement), intersection of A013929 and A000037.

Related to A177712 via A225546.

a051144 n = a051144_list !! (n-1)
a051144_list = filter ((== 0) . a008966) a000037_list
-- Reinhard Zumkeller, Sep 02 2013, Jan 24 2013

[k:k in [1..200]| not IsSquare(k) and not IsSquarefree(k)]; // Marius A. Burtea, Dec 29 2019

N:= 10000;  # to get all entries up to N
A051144:= remove(numtheory:-issqrfree,{$1..N}) minus {seq(i^2,i=1..floor(sqrt(N)))}:
# Robert Israel, Mar 30 2014

searchMax = 32; Complement[Select[Range[searchMax^2], MoebiusMu[#] == 0 &], Range[searchMax]^2] (* Alonso del Arte, Dec 20 2019 *)

is(n)=!issquare(n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A051144(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+(y:=isqrt(x))+sum(mobius(k)*(x//k**2) for k in range(1, y+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 23 2025

(1 - A008966(a(n)))*(1 - A010052(a(n))) = 1; A008966(a(n)) + A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 24 2013

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(2*s) - zeta(s)/zeta(2*s), for s > 1. - Amiram Eldar, Dec 03 2022

Incorrect comment removed by Charles R Greathouse IV, Mar 19 2010

Offset corrected by Reinhard Zumkeller, Jan 24 2013

A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset: 1

Waldemar Puszkarz, Mar 19 2023

nonn

Appears to be a supersequence of A210490, and so also of positive squares and squarefree numbers (A005117). The first term that belongs in here but not in A210490 is 48. The nonsquarefree terms that are not squares are of the form p^(4k)*a, where a is a squarefree number, p is prime, and k > 0. About half of perfect numbers are of this form; one example is 496 = 2^4*31. The sequence has an asymptotic density of about 0.6420.

			48 has 3 square divisors (1, 4, and 16) and 7 nonsquare ones. Consequently, gcd(3,7)=1, thus 48 is a term.

Index entries for sequences related to divisors of numbers

Cf. A210490, A005117 (subsequences), A046951 (number of square divisors), A056595 (number of nonsquare divisors).

Select[Range[100],CoprimeQ[Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#]),DivisorSigma[0,#]-Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#])]&]

for(n=1, 100, a=divisors(n); c=0; for(i=1, #a, issquare(a[i])&&c++); gcd(#a-c, c)==1&&print1(n, ", "))

isok(n) = gcd(numdiv(n), numdiv(sqrtint(n/core(n))))==1 \\ Andrew Howroyd, Mar 19 2023

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A340682 The closure under squaring of the nonunit squarefree numbers.

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A051144 Nonsquarefree nonsquares: each term has a square factor but is not a perfect square itself.

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A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

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