Voozh

A364262

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 distinct prime factors as a(n-1) * a(n).

1, 2, 10, 4, 6, 8, 7, 3, 11, 9, 5, 13, 23, 16, 12, 24, 27, 17, 19, 25, 15, 51, 33, 37, 31, 32, 14, 46, 20, 22, 38, 28, 42, 18, 36, 30, 40, 26, 34, 44, 58, 29, 43, 35, 55, 47, 41, 53, 52, 50, 60, 45, 21, 39, 63, 49, 59, 64, 48, 57, 69, 61, 65, 67, 79, 73, 71, 81, 54, 56, 70, 80, 74, 76, 62, 68, 72

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OFFSET

1,2

COMMENTS

The terms are concentrated along the line a(n) = n, resulting in seven-hundred six fixed points in the first 50000 terms. These begin 1, 2, 4, 7, 19, 43, 50, 134, ... . See the linked image. In the same range the smallest unseen number is 46410, 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 distinct prime factor.

a(3) = 10 as a(2) + 10 = 2 + 10 = 12 while a(2) * 10 = 2 * 10 = 20, both of which have two distinct prime factors.

MATHEMATICA

nn = 120;

c[_] := False; f[x_] := PrimeNu[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}];