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

1,1

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.

Links

Table of n, a(n) for n=1..78.

Index entries for sequences related to prime signature

Example

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.

Mathematica

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

Prog

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

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.

Sequence in context: A131511 A361634 A210490 * A166155 A342525 A325457

Adjacent sequences: A340679 A340680 A340681 * A340683 A340684 A340685

Keyword

nonn,easy

Author

Antti Karttunen and Peter Munn, Feb 07 2021

Status

approved