%I #12 Jul 23 2022 09:54:27

%S 1,2,3,5,8,7,4,11,6,13,17,12,19,23,18,29,31,16,37,41,20,43,27,28,9,47,

%T 24,53,10,30,36,42,44,14,15,59,21,32,61,22,67,71,45,50,25,52,26,63,73,

%U 40,79,33,48,54,66,72,68,56,70,60,75,81,84,76,64,88,90,34,78,80,35,38,83,39,46,49,51

%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has the same number of prime divisors as the sum a(n-2) + a(n-1).

%C In the first 500000 terms on seventeen occasions the sum of the previous two terms equals the next term, these terms being 3, 5, 8, 11, 100,... ,131072, 262144. It in unknown if there are infinitely many such terms. In the same range there are seventy-three fixed points; see A356017. The sequence is conjectured to be a permutation of the positive integers.

%H Scott R. Shannon, <a href="/A355702/a355702_1.png">Image of the first 500000 terms</a>. The green line is y = n.

%e a(4) = 5 as a(2) + a(3) = 2 + 3 = 5 which has one prime divisor, and 5 is the smallest unused number that has one prime divisor.

%e a(6) = 7 as a(4) + a(5) = 5 + 8 = 13 which has one prime divisor, and 7 is the smallest unused number that has one prime divisor.

%e a(7) = 4 as a(5) + a(6) = 8 + 7 = 15 which has two prime divisors, and 4 is the smallest unused number that has two prime divisors.

%Y Cf. A356017, A001222, A355647, A355649, A352867.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jul 14 2022