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.

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

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