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%I #9 Oct 05 2021 11:53:48
%S 1,2,2,2,3,3,2,3,3,3,4,3,4,3,4,4,3,4,3,3,5,3,4,5,5,4,5,5,4,4,6,5,5,4,
%T 4,5,5,6,5,5,4,7,6,6,7,6,6,5,6,5,7,5,6,7,7,7,8,7,7,6,6,6,7,7,8,6,6,7,
%U 7,7,7,8,7,8,8,7,7,8,6,6,8,8,8,7,5,9
%N Number of summands in s-greedy sum of s(n), where s(n) = A000041(n), the partitions numbers.
%C See A242252 for the definitions of greedy sum and summability.
%C Conjecture: A000041(n) is A000041-greedy summable for n >= 3.
%H Clark Kimberling, <a href="/A242258/b242258.txt">Table of n, a(n) for n = 2..1000</a>
%e n... s(n) .. a(n) .... s-greedy sum for s(n)
%e 1 ... 1 .... (none).. (undefined)
%e 2 ... 2 .... 1 ....... 1
%e 3 ... 3 .... 2 ....... 2 + 1
%e 4 ... 5 .... 2 ....... 3 + 2
%e 5 ... 7 .... 2 ....... 5 + 2
%e 6 ... 11 ... 3 ....... 7 + 3 + 1
%e 7 ... 15 ... 3 ....... 11 + 3 + 1
%e 8 ... 22 ... 2 ....... 15 + 7
%e 9 ... 30 ... 3 ....... 22 + 7 + 1
%e 10 .. 42 ... 3 ....... 30 + 11 + 1
%e 11 .. 56.... 4 ....... 42 + 11 + 3
%e 12 .. 77.... 3 ....... 56 + 11 + 5 + 1
%t z = 200; s = Table[PartitionsP[n], {n, 1, z}]; t = Table[{s[[n]], #, Total[#] == s[[n]]} &[DeleteCases[-Differences[FoldList[If[#1 - #2 >= 0, #1 - #2, #1] &, s[[n]], Reverse[Select[s, # < s[[n]] &]]]], 0]], {n, z}]; r[n_] := s[[n]] - Total[t[[n]][[2]]]; c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* _Peter J. C. Moses_, May 06 2014 *)
%Y Cf. A242259, A241833, A242252, A000041.
%K nonn,easy
%O 2,2
%A _Clark Kimberling_, May 10 2014