cp's OEIS Frontend

A356851 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest positive number not occurring earlier such that a(n) shares a factor with the previous Omega(a(n)) terms.

%I A356851 #18 May 07 2023 19:33:09
%S A356851 1,2,4,6,3,9,12,15,5,10,20,14,7,21,28,35,30,25,40,45,50,18,22,8,16,24,
%T A356851 26,13,39,52,65,78,60,33,11,44,55,66,70,34,17,51,68,85,102,90,38,19,
%U A356851 57,76,95,114,110,46,23,69,92,115,138,130,58,29,87,116,145,174,150,62,31,93,124,155,186
%N A356851 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest positive number not occurring earlier such that a(n) shares a factor with the previous Omega(a(n)) terms.
%C A356851 For n > 250 the terms are concentrated along seven lines, see the linked images. Unlike the other six lines, numbers along the second lowest line are somewhat spread out, and these terms contain all numbers with Omega(a(n)) > 1. The lowest line contains all the primes, while the upper five lines contain terms with Omega(a(n)) = 2, 3, and 4. The primes up to n=100000 occur in their natural order except for 11 and 13 which are switched. The only fixed point beyond the first two terms is 10, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
%H A356851 Scott R. Shannon, <a href="/A356851/a356851.png">Image of n=1..100000</a>. The green line is a(n) = n.
%H A356851 Scott R. Shannon, <a href="/A356851/a356851_1.png">Image of n=1..100000 with color</a>. Terms with Omega(a(n)) = 1, 2, 3, 4, 5, >=6 are shown in red, yellow, green, blue, violet, white respectively.
%e A356851 a(8) = 15 as Omega(15) = A001222(15) = 2, and 15 shares a factor with the previous two terms, namely a(6) = 9 and a(7) = 12.
%Y A356851 Cf. A064413, A356850 (coprime to previous Omega(a(n)) terms), A001222, A093714, A336957, A000040
%K A356851 nonn
%O A356851 1,2
%A A356851 _Scott R. Shannon_, Aug 31 2022