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A274011 a(n) is the greatest number of elements in a partition of n into distinct parts such that no two elements add to another. 0
1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A lower bound for a(n^2) is n (use the n^2 partition of the first n consecutive odd numbers.)

An upper bound u for a(n) is found by a partition of A057944(n); [1,..,u]. This gives u = floor((floor(sqrt(8*n+7))-1)/2). - David A. Corneth, Jun 06 2016

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

a(24) = 5 because {1,2,4,7,10} is a partition of 24 and there are no sum-free partitions with more parts.

Candidates for such a partition of size 5 of 24 are found by adding [0,1,2,3,4] to partitions of 5 of 24 - (0+1+2+3+4). - David A. Corneth, Jun 06 2016

a(25) = 5 because {1,3,5,7,9} is a partition of 25. {1,2,4,7,11} does not show that a(25) >= 5 because 4,7, and 11 are all elements of the set and 4+7=11.

MATHEMATICA

dif[w_] := Length[w] <= 2 || {} == Intersection[w, Reap[ Do[ Sow[w[[i]] + w[[j]]], {i, Length@ w}, {j, i-1}]][[2, 1]]]; p[tg_, w_] := If[tg == 0, bst = Max[bst, Length@ w], Block[{v=If[w == {}, 0, Last@w], u}, Do[u = Append[w, k]; If[dif@ u, p[tg-k, u]], {k, v+1, tg}]]]; a[n_] := (bst = 0; p[n, {}]; bst); Array[a, 50] (* Giovanni Resta, Jun 06 2016 *)

CROSSREFS

Sequence in context: A090973 A076634 A083277 * A307726 A235123 A242768

Adjacent sequences:  A274008 A274009 A274010 * A274012 A274013 A274014

KEYWORD

nonn

AUTHOR

Gordon Hamilton, Jun 06 2016

EXTENSIONS

a(25)-a(86) from Giovanni Resta, Jun 06 2016

STATUS

approved

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Last modified August 8 08:27 EDT 2020. Contains 336293 sequences. (Running on oeis4.)