A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 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, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset

1,1

Comments

Numbers not of the form s^(2^e), where s is a squarefree number, and e >= 0.

The categorization provided by this sequence and its complement, A340682, 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 more than one term, that is numbers m such that A331591(m) > 1.

Presence in the sequence is determined by prime signature (A101296). The set of represented signatures starts: {{3}, {2,1}, {3,1}, {2,1,1}, {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,1,1}, {2,1,1,1,1}, {7}, ...}.

Gives positions of 1's in A340675 after its initial one.

Links

Table of n, a(n) for n=1..70.

Index entries for sequences related to prime signature

Example

24 = 6 * 4 = 6^1 * 2^2 = 6^(2^0) * 2^(2^1), which is the factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 24 is in the sequence.
The equivalent factorization for 100 is 100 = 10^2 = 10^(2^1). As this factorization has only 1 term, 100 is not in the sequence.

Prog

(PARI) isA340681(n) = if(!issquare(n), !issquarefree(n), (n>1)&&isA340681(sqrtint(n)));

Crossrefs

Cf. A101296, A331591, A340675.

Cf. A340682 (complement, apart from 1 which is in neither).

Cf. subsequences: A051144, A059404.

Subsequence of A013929.

Sequence in context: A352880 A104201 A036327 * A051144 A067537 A046339

Adjacent sequences: A340678 A340679 A340680 * A340682 A340683 A340684

Keyword

nonn,easy

Author

Antti Karttunen and Peter Munn, Feb 07 2021

Status

approved