A323420 Lexicographically earliest sequence of positive integers such that for any n > 0, a(n + a(n)) > a(n).

1, 2, 1, 3, 1, 2, 4, 3, 1, 2, 5, 3, 1, 2, 4, 6, 1, 2, 5, 3, 1, 7, 4, 6, 1, 2, 5, 3, 8, 7, 4, 6, 1, 2, 5, 3, 9, 7, 4, 6, 1, 2, 5, 3, 8, 10, 4, 6, 1, 2, 5, 3, 9, 7, 4, 11, 1, 2, 5, 3, 8, 10, 4, 6, 1, 2, 12, 3, 9, 7, 4, 11, 1, 2, 5, 3, 8, 10, 13, 6, 1, 2, 12, 3

Offset: 1

Author

Rémy Sigrist, Aug 30 2019

Keywords

• nonn

Comments

Every positive integer appears in the sequence.

Empirically:

- for any n > 0, the least d > 0 such that a(n) = a(n+d) is a power of 2 (see scatterplot in Links section),

- the run-length transform of the first differences of the positions of the 1's in the sequence corresponds to A055010 (excluding the leading 0).

Examples

			a(1) = 1, hence a(1 + a(1)) = a(2) > 1.
a(2) = 2, hence a(2 + a(2)) = a(4) > 2.
a(3) = 1, hence a(3 + a(1)) = a(4) > 1.
a(4) = 3, etc.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of the least d > 0 such that a(n) = a(n+d))

Crossrefs

Cf. A000124, A055010.

Programs

PARI

a = vector(84, n, 1); for (n=1, #a, print1 (a[n] ", "); nan = n+a[n]; if (nan <= #a, a[nan] = max(a[nan], 1+a[n])))

Formula

a(A000124(n)) = n + 1 for any n >= 0.