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.
With n = 10, the maximum value of m is 9 and all values of m from 1 to 9 appear within the first 10 rows of A008284, so a(10) = 0. With n = 13, the maximum value of m is 18 and all values of m from 1 to 18, except 16 and 17, appear within the first 13 rows of A008284, so a(13) = 2.
FoldList[{#1, Count[Range@ Max@ #2, k_ /; FreeQ[#1, k]]} & @@ {Append[#1, #2], #2} & @@ {First@ #1, Table[SeriesCoefficient[1/QPochhammer[a q, q], {q, 0, #2}, {a, 0, k}], {k, #2}]} &, {{{1}}, 0}, Range[2, 40]][[All, -1]] (* Michael De Vlieger, Sep 29 2017, after Vladimir Reshetnikov at A008284 *)
For n=9 the counts of partitions for k=1..9 is 0,0,1,4,5,3,2,1,1 so the maximum is 5 (at k=5).
b:= proc(n, i, t) option remember; `if`(n=0, 1,
`if`(i*t max(seq(b(n-i, min(n-i, i-1), i), i=0..n)):
seq(a(n), n=0..55); # Alois P. Heinz, May 12 2022
Max @@@ Table[T[n,k,k]-T[n,k-1,k],{n,1,128},{k,n}] (* with T[n,a,b] as defined in A047993 *)
f[n_] := Block[{k = 1, mk = mx = 0}, While[k < n + 1, a = Length@ IntegerPartitions[n, {k}]; If[a > mx, mx = a; mk = k]; k++ ]; mk]; Array[f, 85] (* Robert G. Wilson v, Jul 20 2010 *)
The sequence of terms together with their prime indices begins:
2: {1}
2: {1}
4: {1,1}
8: {1,1,1}
24: {1,1,1,2}
72: {1,1,1,2,2}
600: {1,1,1,2,3,3}
4200: {1,1,1,2,3,3,4}
101640: {1,1,1,2,3,4,5,5}
2042040: {1,1,1,2,3,4,5,6,7}
107869080: {1,1,1,2,3,5,5,7,8,9}
6435365640: {1,1,1,2,3,5,5,7,10,10,11}
644779672680: {1,1,1,2,3,5,6,7,11,12,13,15}
Times@@@Table[If[n>0&&k==0,1,Prime[Length[IntegerPartitions[n,{k}]]]],{n,0,20},{k,0,n}]
f[n_] := Block[{k = 1, mk = mx = 0}, While[k < n + 1, a = Length@ IntegerPartitions[n, {k}]; If[a > mx, mx = a; mk = k]; k++ ]; mk]; Array[f, 85] (* Robert G. Wilson v, Jul 20 2010 *)
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