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