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A288164 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+2) has at least 5 distinct prime factors.

%I A288164 #17 Jun 16 2017 22:21:53
%S A288164 1,2,2310,1155,3,4,770,1365,6,8,385,1785,12,10,455,231,18,20,595,273,
%T A288164 22,30,105,77,26,60,165,91,14,66,195,35,28,78,255,55,38,42,210,65,11,
%U A288164 84,390,85,7,114,330,70,13,33,420,130,17,21,462,110,5,39,546,140
%N A288164 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+2) has at least 5 distinct prime factors.
%C A288164 This sequence is a permutation of the natural numbers, with inverse A288799.
%C A288164 Conjecturally, a(n) ~ n.
%C A288164 For k >= 0, let f_k be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+k) has at least 5 distinct prime factors.
%C A288164 In particular, we have:
%C A288164 - f_0 = the numbers with at least 5 distinct prime factors,
%C A288164 - f_1 = A285487,
%C A288164 - f_2 = a (this sequence),
%C A288164 - f_3 = A288171.
%C A288164 If k > 0, then:
%C A288164 - f_k is a permutation of the natural numbers,
%C A288164 - f_k(i) = i for any i <= k,
%C A288164 - f_k(k+1) = A002110(5),
%C A288164 - conjecturally, f_k(n) ~ n.
%H A288164 Rémy Sigrist, <a href="/A288164/b288164.txt">Table of n, a(n) for n = 1..25000</a>
%H A288164 Rémy Sigrist, <a href="/A288164/a288164.txt">C++ program for A288164</a>
%H A288164 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A288164 The first terms, alongside the primes p dividing a(n)*a(n+2), are:
%e A288164 n       a(n)    p
%e A288164 --      ----    --------------
%e A288164 1       1       2, 3, 5, 7, 11
%e A288164 2       2       2, 3, 5, 7, 11
%e A288164 3       2310    2, 3, 5, 7, 11
%e A288164 4       1155    2, 3, 5, 7, 11
%e A288164 5       3       2, 3, 5, 7, 11
%e A288164 6       4       2, 3, 5, 7,     13
%e A288164 7       770     2, 3, 5, 7, 11
%e A288164 8       1365    2, 3, 5, 7,     13
%e A288164 9       6       2, 3, 5, 7, 11
%e A288164 10      8       2, 3, 5, 7,         17
%e A288164 11      385     2, 3, 5, 7, 11
%e A288164 12      1785    2, 3, 5, 7,         17
%e A288164 13      12      2, 3, 5, 7,     13
%e A288164 14      10      2, 3, 5, 7, 11
%e A288164 15      455     2, 3, 5, 7,     13
%e A288164 16      231     2, 3, 5, 7, 11
%e A288164 17      18      2, 3, 5, 7,         17
%e A288164 18      20      2, 3, 5, 7,     13
%e A288164 19      595     2,    5, 7, 11,     17
%e A288164 20      273     2, 3, 5, 7,     13
%e A288164 21      22      2, 3, 5, 7, 11
%e A288164 22      30      2, 3, 5, 7, 11
%e A288164 23      105     2, 3, 5, 7,     13
%Y A288164 Cf. A002110, A285487, A288171, A288799 (inverse).
%K A288164 nonn,look
%O A288164 1,2
%A A288164 _Rémy Sigrist_, Jun 16 2017