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%I #16 Feb 13 2021 07:45:53
%S 8,12,18,20,24,27,28,32,40,44,45,48,50,52,54,56,60,63,64,68,72,75,76,
%T 80,84,88,90,92,96,98,99,104,108,112,116,117,120,124,125,126,128,132,
%U 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
%N The closure under squaring of A051144, the nonsquarefree nonsquares.
%C Numbers not of the form s^(2^e), where s is a squarefree number, and e >= 0.
%C 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.
%C 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.
%C 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}, ...}.
%C Gives positions of 1's in A340675 after its initial one.
%H <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%e 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.
%e 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.
%o (PARI) isA340681(n) = if(!issquare(n), !issquarefree(n), (n>1)&&isA340681(sqrtint(n)));
%Y Cf. A101296, A331591, A340675.
%Y Cf. A340682 (complement, apart from 1 which is in neither).
%Y Cf. subsequences: A051144, A059404.
%Y Subsequence of A013929.
%K nonn,easy
%O 1,1
%A _Antti Karttunen_ and _Peter Munn_, Feb 07 2021
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