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

1,2

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.

Links

Table of n, a(n) for n=1..90.

Example

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.

Mathematica

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

Crossrefs

Cf. A000041, A080577.

Sequence in context: A035375 A354143 A358616 * A093493 A087162 A288914

Adjacent sequences: A366102 A366103 A366104 * A366106 A366107 A366108

Keyword

nonn

Author

Clark Kimberling, Oct 03 2023

Status

approved