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%I #21 Jun 14 2017 02:46:49
%S 1,4,2,6,3,8,5,9,7,12,10,14,16,11,18,13,20,15,21,24,17,25,19,27,22,26,
%T 28,23,32,29,36,30,33,39,40,31,44,34,38,42,35,45,37,48,41,49,43,50,46,
%U 52,47,54,51,56,53,60,55,63,57,64,58,62,66,68,59,72,61
%N Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.
%C The sequence can always be extended with a number that is not squarefree (say a multiple of 4); after a term that is not squarefree, we can extend the sequence with the least unused number; as there are infinitely many multiples of 4, this sequence is a permutation of the natural numbers (with inverse A285297).
%C Conjecturally, a(n) ~ n.
%C This sequence has similarities with A075380: here we consider the product of consecutive terms, there the sum of consecutive terms.
%C For any k>0, let b_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^k for some prime p; in particular we have:
%C - b_1 = A000027 (the natural numbers),
%C - b_2 = a (this sequence),
%C - b_3 = A285299,
%C - b_4 = A285386,
%C - b_5 = A285417.
%C For any k>0, b_k is a permutation of the natural numbers.
%C For any k>0, b_k(1)=1 and b_k(2)=2^k.
%C Graphically, the sequences from b_2 to b_5 differ.
%H Rémy Sigrist, <a href="/A285296/b285296.txt">Table of n, a(n) for n = 1..2000</a>
%H Rémy Sigrist, <a href="/A285296/a285296.gp.txt">PARI program for A285296</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e The first terms, alongside the primes p such that p^2 divides a(n)*a(n+1), are:
%e n a(n) p
%e -- ---- -
%e 1 1 2
%e 2 4 2
%e 3 2 2
%e 4 6 3
%e 5 3 2
%e 6 8 2
%e 7 5 3
%e 8 9 3
%e 9 7 2
%e 10 12 2
%e 11 10 2
%e 12 14 2
%e 13 16 2
%e 14 11 3
%e 15 18 3
%e 16 13 2
%e 17 20 2, 5
%e 18 15 3
%e 19 21 2, 3
%e 20 24 2
%Y Cf. A000027, A075380, A285297 (inverse).
%K nonn
%O 1,2
%A _Rémy Sigrist_, Apr 16 2017
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