A363162 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 distinct prime divisors as a(n-2) + a(n-1).

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 14, 15, 9, 18, 11, 13, 20, 21, 16, 17, 22, 24, 26, 28, 33, 19, 34, 23, 35, 36, 25, 27, 38, 39, 40, 29, 44, 31, 45, 46, 48, 50, 51, 32, 37, 52, 41, 54, 55, 43, 56, 57, 47, 58, 30, 62, 63, 49, 65, 42, 53, 68, 59, 61, 60, 64, 69, 72, 74, 75, 67, 76, 77, 80, 71, 73, 82

Offset: 1

Author

Scott R. Shannon, Jul 06 2023

Keywords

• nonn

Comments

The terms with different numbers of distinct prime divisors are concentrated along different lines in the graph; see the attached colored image. There are numerous fixed points in the first one million terms, although the last nonprime fixed point is a(n) = 3495. Beyond that there are thirty-one more fixed points all with prime values; it is likely more exist although this is unknown. The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(4) = 4 as a(2) + a(3) = 2 + 3 = 5 which has one distinct prime divisor, and 4 is the smallest unused number that has one distinct prime divisor.
a(10) = 12 as a(8) + a(9) = 8 + 10 = 18 which has two distinct prime divisors, and 12 is the smallest unused number that has two distinct prime divisors.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Colored image of the first one million terms. The terms with 1,2,3,4,5,6 or 7 distinct prime divisors are colored across the spectrum from red to violet respectively. The white line is a(n) = n.

Crossrefs

Cf. A001221, A355702, A355647, A355649, A352867.

Programs

Mathematica

nn = 120; c[] := False; f[x] := PrimeNu[x]; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; i = 1; j = s = 2; u = 3; Do[k = u; s = f[i + j]; While[Or[c[k], f[k] != s], k++]; Set[{a[n], c[k], i, j}, {k, True, j, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jul 08 2023 *)