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A242258
Number of summands in s-greedy sum of s(n), where s(n) = A000041(n), the partitions numbers.
2
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, 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, 7, 7, 7, 8, 7, 8, 8, 7, 7, 8, 6, 6, 8, 8, 8, 7, 5, 9
OFFSET
2,2
COMMENTS
See A242252 for the definitions of greedy sum and summability.
Conjecture: A000041(n) is A000041-greedy summable for n >= 3.
LINKS
EXAMPLE
n... s(n) .. a(n) .... s-greedy sum for s(n)
1 ... 1 .... (none).. (undefined)
2 ... 2 .... 1 ....... 1
3 ... 3 .... 2 ....... 2 + 1
4 ... 5 .... 2 ....... 3 + 2
5 ... 7 .... 2 ....... 5 + 2
6 ... 11 ... 3 ....... 7 + 3 + 1
7 ... 15 ... 3 ....... 11 + 3 + 1
8 ... 22 ... 2 ....... 15 + 7
9 ... 30 ... 3 ....... 22 + 7 + 1
10 .. 42 ... 3 ....... 30 + 11 + 1
11 .. 56.... 4 ....... 42 + 11 + 3
12 .. 77.... 3 ....... 56 + 11 + 5 + 1
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 10 2014
STATUS
approved