OFFSET

1,2

COMMENTS

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.

LINKS

Scott R. Shannon, Table of n, a(n) for n = 1..10000.

Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n.

EXAMPLE

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

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

MATHEMATICA

nn = 120;

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

c[1] = True; u = 2;

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

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

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

Array[a, nn] (* Michael De Vlieger, Jul 17 2023 *)

CROSSREFS

KEYWORD

nonn

AUTHOR

Scott R. Shannon, Jul 16 2023

STATUS

approved