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%I #15 Jun 14 2017 02:47:02
%S 1,30030,2,15015,4,19635,6,5005,12,6545,18,7315,24,7735,22,1365,34,
%T 1155,26,1785,38,2145,14,2805,28,3135,42,715,84,935,78,385,102,455,66,
%U 595,114,770,39,1190,33,910,51,1330,69,1430,21,1870,57,1540,87,1610,93
%N Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 distinct prime factors.
%C This sequence can always be extended with a multiple of 30030 = 2*3*5*7*11*13; after a term that has at least 6 distinct prime factors, we can extend the sequence with the least unused number; as there are infinitely many numbers with at least 6 distinct prime factors, this sequence is a permutation of the natural numbers (with inverse A285656).
%C Conjecturally, a(n) ~ n.
%C The first fixed points are: 1, 39, 1344, 1350, 3556, 3560, 5738, 6974, 15668585, 15668673, 15668787.
%C For any k>0, let d_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms has at least k distinct prime factors; in particular we have:
%C - d_1 = A000027 (the natural numbers),
%C - d_5 = A285487,
%C - d_6 = a (this sequence).
%C For any k>0:
%C - d_k is a permutation of the natural numbers,
%C - d_k(1) = 1 and d_k(2) = A002110(k),
%C - conjecturally: d_k(n) ~ n.
%H Rémy Sigrist, <a href="/A285655/b285655.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A285655/a285655.png">Scatterplot of the first 20 000 000 terms</a>
%H Rémy Sigrist, <a href="/A285655/a285655_1.png">Scatterplot of the first difference of the first 20 000 000 terms</a>
%H Rémy Sigrist, <a href="/A285655/a285655.txt">C++ program for A285655</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 dividing a(n)*a(n+1), are:
%e n a(n) p
%e -- ---- ------------------
%e 1 1 2, 3, 5, 7, 11, 13
%e 2 30030 2, 3, 5, 7, 11, 13
%e 3 2 2, 3, 5, 7, 11, 13
%e 4 15015 2, 3, 5, 7, 11, 13
%e 5 4 2, 3, 5, 7, 11, 17
%e 6 19635 2, 3, 5, 7, 11, 17
%e 7 6 2, 3, 5, 7, 11, 13
%e 8 5005 2, 3, 5, 7, 11, 13
%e 9 12 2, 3, 5, 7, 11, 17
%e 10 6545 2, 3, 5, 7, 11, 17
%e 11 18 2, 3, 5, 7, 11, 19
%e 12 7315 2, 3, 5, 7, 11, 19
%e 13 24 2, 3, 5, 7, 13, 17
%e 14 7735 2, 5, 7, 11, 13, 17
%e 15 22 2, 3, 5, 7, 11, 13
%e 16 1365 2, 3, 5, 7, 13, 17
%e 17 34 2, 3, 5, 7, 11, 17
%e 18 1155 2, 3, 5, 7, 11, 13
%e 19 26 2, 3, 5, 7, 13, 17
%e 20 1785 2, 3, 5, 7, 17, 19
%Y Cf. A000027, A002110, A285487, A285656 (inverse).
%K nonn
%O 1,2
%A _Rémy Sigrist_, Apr 23 2017
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