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%I #4 May 15 2014 10:14:52
%S 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,3,3,2,3,3,3,3,3,3,2,4,4,
%T 3,2,4,4,3,3,4,4,3,2,4,4,4,3,4,4,4,4,5,4,4,3,3,4,5,4,4,4,4,4,4,3,5,4,
%U 5,5,4,4,5,3,5,4,5,5,5,5,4,5,4,3,4,4
%N Number of summands in s-greedy sum of s(n), where s(n) = A000009(n) (strict partition numbers).
%C See A242252 for the definitions of greedy sum and summability. Conjecture: A000009(n) is A000009-greedy summable for n >= 3.
%C n... s(n) .... a(n) .... s-greedy sum for s(n)
%C 1 ... 1 ...... (undefined)
%C 2 ... 2 ...... (undefined)
%C 3 ... 2 ...... 2 ........ 1 + 1
%C 4 ... 2 .......2 ........ 1 + 1
%C 5 ... 3 .......2 ....... 2 + 1
%C 6 ... 4 ...... 2 ........ 3 + 1
%C 7 ... 5 ...... 2 ........ 4 + 1
%C 8 ... 6 ...... 2 ........ 5 + 1
%C 9 ... 8 ...... 2 ........ 6 + 2
%C 10 .. 10 ..... 2 ........ 8 + 2
%C 25 .. 142 .... 3 ........ 122 + 18 + 2
%C 35 .. 585 .... 4 ........ 512 + 64 + 8 + 1
%C 55 .. 6378 ... 5 ........ 5718 + 585 + 64 + 10 + 1
%H Clark Kimberling, <a href="/A242259/b242259.txt">Table of n, a(n) for n = 3..999</a>
%e n... s(n) .... a(n) .... s-greedy sum for s(n)
%e 1 ... 1 ...... (undefined)
%e 2 ... 2 ...... (undefined)
%e 3 ... 2 ...... 2 ........ 1 + 1
%e 4 ... 2 .......2 ........ 1 + 1
%e 5 ... 3 .......2 ....... 2 + 1
%e 6 ... 4 ...... 2 ........ 3 + 1
%e 7 ... 5 ...... 2 ........ 4 + 1
%e 8 ... 6 ...... 2 ........ 5 + 1
%e 9 ... 8 ...... 2 ........ 6 + 2
%e 10 .. 10 ..... 2 ........ 8 + 2
%e 25 .. 142 .... 3 ........ 122 + 18 + 2
%e 35 .. 585 .... 4 ........ 512 + 64 + 8 + 1
%e 55 .. 6378 ... 5 ........ 5718 + 585 + 64 + 10 + 1
%t z = 200; s = Table[PartitionsQ[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. A242258, A241833, A242252, A000009.
%K nonn,easy
%O 3,1
%A _Clark Kimberling_, May 10 2014
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