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A242259
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Number of summands in s-greedy sum of s(n), where s(n) = A000009(n) (strict partition numbers).
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2
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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, 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, 5, 5, 4, 4, 5, 3, 5, 4, 5, 5, 5, 5, 4, 5, 4, 3, 4, 4
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OFFSET
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3,1
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COMMENTS
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See A242252 for the definitions of greedy sum and summability. Conjecture: A000009(n) is A000009-greedy summable for n >= 3.
n... s(n) .... a(n) .... s-greedy sum for s(n)
1 ... 1 ...... (undefined)
2 ... 2 ...... (undefined)
3 ... 2 ...... 2 ........ 1 + 1
4 ... 2 .......2 ........ 1 + 1
5 ... 3 .......2 ....... 2 + 1
6 ... 4 ...... 2 ........ 3 + 1
7 ... 5 ...... 2 ........ 4 + 1
8 ... 6 ...... 2 ........ 5 + 1
9 ... 8 ...... 2 ........ 6 + 2
10 .. 10 ..... 2 ........ 8 + 2
25 .. 142 .... 3 ........ 122 + 18 + 2
35 .. 585 .... 4 ........ 512 + 64 + 8 + 1
55 .. 6378 ... 5 ........ 5718 + 585 + 64 + 10 + 1
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LINKS
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EXAMPLE
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n... s(n) .... a(n) .... s-greedy sum for s(n)
1 ... 1 ...... (undefined)
2 ... 2 ...... (undefined)
3 ... 2 ...... 2 ........ 1 + 1
4 ... 2 .......2 ........ 1 + 1
5 ... 3 .......2 ....... 2 + 1
6 ... 4 ...... 2 ........ 3 + 1
7 ... 5 ...... 2 ........ 4 + 1
8 ... 6 ...... 2 ........ 5 + 1
9 ... 8 ...... 2 ........ 6 + 2
10 .. 10 ..... 2 ........ 8 + 2
25 .. 142 .... 3 ........ 122 + 18 + 2
35 .. 585 .... 4 ........ 512 + 64 + 8 + 1
55 .. 6378 ... 5 ........ 5718 + 585 + 64 + 10 + 1
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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