A366105 - OEIS

%I #21 Oct 06 2023 11:07:47

%S 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,

%T 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,

%U 3,3,4,3,4,5,3,4,4,5,6,4,4,5,5,6,7,4,5,6,5,6

%N 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).

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

%C 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.

%e 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.

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

%Y Cf. A000041, A080577.

%K nonn

%O 1,2

%A _Clark Kimberling_, Oct 03 2023