A366105 a(n) is the number of parts in the n-th partition of n when the partitions are listed in graded reverse lexicographic order (cf. A080577, as in Mathematica).

1, 2, 3, 3, 3, 3, 4, 2, 3, 3, 4, 5, 2, 3, 3, 4, 4, 5, 6, 2, 3, 3, 4, 3, 4, 5, 4, 5, 6, 7, 2, 3, 3, 4, 3, 4, 5, 4, 4, 5, 6, 5, 6, 7, 8, 2, 3, 3, 4, 3, 4, 5, 3, 4, 4, 5, 6, 4, 5, 5, 6, 7, 5, 6, 7, 8, 9, 2, 3, 3, 4, 3, 4, 5, 3, 4, 4, 5, 6, 4, 4, 5, 5, 6, 7, 4, 5, 6, 5, 6

Offset: 1

Author

Clark Kimberling, Oct 03 2023

Keywords

• nonn

Comments

Conjecture 1. Every integer m > 1 occurs infinitely many times. (For example, 2 occurs for n = 2,8,13,20,31,46,68,....)

Conjecture 2. Let f(n) be the greatest (i.e., the first) part in the n-th partition of n. Then for every integer m, there exists an index i such that f(i+1), f(i+2), ..., f(i+m) are consecutive integers.

Examples

			The partitions of 5, listed in reverse-lexicographic order, are (5, 41, 32, 311, 221, 2111, 11111); the 5th in this list is 221, with length 3, so that a(5) = 3.
		

Crossrefs

Cf. A000041, A080577.

Programs

Mathematica

Table[Length[IntegerPartitions[n][[n]]], {n, 1, 40}]