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

Author

Antti Karttunen and Peter Munn, Feb 07 2021

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• Index entries for sequences related to prime signature

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Python

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