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%I #22 Oct 29 2018 05:13:43
%S 0,1,2,4,5,6,8,9,10,12,13,14,16,17,18,20,21,22,24,25,26,27,28,29,30
%N Fertility numbers.
%C The fertility of a permutation pi is |s^{-1}(pi)|, where s is West's stack-sorting map. A nonnegative integer is called a fertility number if it is the fertility of some permutation.
%C The set of fertility numbers is closed under multiplication.
%C Every nonnegative integer that is not congruent to 3 modulo 4 is a fertility number.
%C The lower asymptotic density of this sequence is at least 0.7618. In particular, there are infinitely many fertility numbers that are congruent to 3 modulo 4. The smallest of these is 27. It appears as though 95 is the second-smallest fertility number that is congruent to 3 modulo 4.
%C It is conjectured that there are infinitely many positive integers that are not fertility numbers.
%C Empirically found 149 terms congruent 3 mod 4, the second smallest being 39 followed by 51, 63, 95, 123, ... - _Jon Maiga_, Oct 28 2018
%H C. Defant, <a href="https://arxiv.org/abs/1809.04421">Fertility numbers</a>, arXiv:1809:04421 [math.CO], 2018.
%H J. Maiga, <a href="http://jonkagstrom.com/fertility-numbers">Stack sorting and fertility numbers</a>, 2018.
%e The preimages of 123 under the stack-sorting map are 123, 132, 213, 312, and 321. This shows that the fertility of 123 is 5, so 5 is a fertility number.
%K nonn,more
%O 1,3
%A _Colin Defant_, Sep 14 2018
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