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%I #14 Feb 26 2017 11:59:08
%S 1,2,3,6,4,5,10,15,7,14,21,28,35,42,8,9,11,22,33,44,12,13,26,39,52,65,
%T 16,17,34,51,68,85,102,119,136,153,170,18,19,38,57,76,95,114,133,152,
%U 171,190,209,228,247,266,285,20,23,46,69,92,115,138,161,24,25
%N Lexicographically earliest sequence of distinct terms such that a(n)=length of first run of multiples of prime(n) in this sequence.
%C For any i>0, let R_i denote the first run of multiples of prime(i) in this sequence; several such runs can overlap.
%C The following table gives the index of the first term belonging to k runs, alongside the corresponding runs, for small values of k:
%C k n Runs
%C -- -------- ----
%C 0 1 None
%C 1 2 R_1
%C 2 1533 R_37 and R_38
%C 3 7693674 R_1534 to R_1536
%C 4 7706584 R_1534 to R_1537
%C 5 7738564 R_1535 to R_1539
%C 6 7751486 R_1535 to R_1540
%C 7 37170235 R_3302 to R_3308
%C 8 43960552 R_3551 to R_3558
%C 9 44051293 R_3552 to R_3560
%C 10 44145862 R_3553 to R_3562
%C 11 44236717 R_3554 to R_3564
%C If i<j, then R_i starts before R_j.
%C For any prime p, there is a multiple of p in this sequence (see A282842).
%C Conjectures:
%C - For any k, there is a term belonging to k runs.
%C - This sequence is a permutation of the natural numbers.
%H Rémy Sigrist, <a href="/A282841/b282841.txt">Table of n, a(n) for n = 1..100000</a>
%H Rémy Sigrist, <a href="/A282841/a282841.png">Logarithmic scatterplot of the first hundred million terms</a>
%H Rémy Sigrist, <a href="/A282841/a282841.cs.txt">C# program for A282841</a>
%e a(1)=1 fits the definition.
%e a(2)=2 fits the definition and introduces R_1; R_1 has length a(1)=1: a(3) must not be a multiple of 2.
%e a(3)=3 fits the definition and the constraint on R_1.
%e a(3) introduces R_2; R_2 has length a(2)=2: a(4) must be a multiple of 3, whereas a(5) must not be a multiple of 3.
%e a(3)=6 fits the definition and the constraint on R_2.
%e a(4)=4 fits the definition and the constraint on R_2.
%e a(5)=5 fits the definition and introduces R_3.
%Y Cf. A282842, A282999.
%K nonn
%O 1,2
%A _Rémy Sigrist_, Feb 22 2017