OFFSET
1,2
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.
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.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 06 2023
STATUS
approved