This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A366105 #21 Oct 06 2023 11:07:47
%S A366105 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 A366105 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 A366105 3,3,4,3,4,5,3,4,4,5,6,4,4,5,5,6,7,4,5,6,5,6
%N 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).
%C A366105 Conjecture 1. Every integer m > 1 occurs infinitely many times. (For example, 2 occurs for n = 2,8,13,20,31,46,68,....)
%C A366105 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 A366105 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 A366105 Table[Length[IntegerPartitions[n][[n]]], {n, 1, 40}]
%Y A366105 Cf. A000041, A080577.
%K A366105 nonn
%O A366105 1,2
%A A366105 _Clark Kimberling_, Oct 03 2023