%I #14 Jul 17 2023 08:50:40

%S 1,2,6,10,14,13,15,3,7,19,9,11,23,4,28,53,5,17,25,35,21,27,29,34,22,

%T 38,37,26,55,33,43,31,39,49,51,41,57,59,45,99,69,47,58,46,66,30,82,71,

%U 77,61,20,44,52,12,148,68,60,196,92,36,220,212,103,62,18,78,122,73,8,127,67,79,74,97,85

%N a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared such that a(n-1) + a(n) has the same number of prime factors as a(n-1) * a(n).

%C The terms are concentrated along a line just above a(n) = n, resulting in twenty-four fixed points in the first 50000 terms. These begin 1, 2, 21, 116, 141, 292, 477, 700, ... . See the linked image. In the same range the smallest unseen number is 342, suggesting all numbers will eventually appear.

%H Scott R. Shannon, <a href="/A364261/b364261.txt">Table of n, a(n) for n = 1..10000</a>.

%H Scott R. Shannon, <a href="/A364261/a364261.png">Image of the first 50000 terms</a>. The green line is a(n) = n.

%e a(2) = 2 as a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one prime factor.

%e a(3) = 6 as a(2) + 6 = 2 + 6 = 8 while a(2) * 6 = 2 * 6 = 12, both of which have three prime factors.

%t nn = 120;

%t c[_] := False; f[x_] := PrimeOmega[x]; a[1] = j = 1;

%t c[1] = True; u = 2;

%t Do[k = u; While[Or[c[k], f[j + k] != f[j k]], k++];

%t Set[{a[n], c[k], j}, {k, True, k}];

%t If[k == u, While[c[u], u++]], {n, 2, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Jul 17 2023 *)

%Y Cf. A364262 (distinct factors), A001222, A027746.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jul 16 2023