%I
%S 1,1,2,1,2,3,2,1,3,4,2,3,2,4,5,1,2,4,2,5,6,4,2,3,5,4,6,7,2,5,2,1,6,4,
%T 7,8,2,4,6,5,2,7,2,8,9,4,2,3,7,5,6,8,2,9,10,7,6,4,2,8,2,4,9,1,10,11,2,
%U 8,6,7,2,9,2,4,10,8,11,12,2,5,9,4,2,8,10,4,6,11,2,12,13,8,6,4,10,3,2,7,11,8,2,12
%N a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
%C n is the sum of at most a(n) consecutive positive integers. As suggested by _David W. Wilson_, Aug 15 2005: Suppose n is to be written as sum of k consecutive integers starting with m, then 2n = k(2m + k  1). Only one of the factors is odd. For each odd divisor d of n there is a unique corresponding k = min(d,2n/d). a(n) is the largest among those k.  Jaap Spies_, Aug 16 2005
%D Nieuw Archief voor Wiskunde 5/6, no. 2, Problems/UWC, Problem C, Jun 2005, pp. 181182.
%H Donovan Johnson, <a href="/A109814/b109814.txt">Table of n, a(n) for n = 1..10000</a>
%H K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath107.htm">Partitions into Consecutive Integers</a>
%H A. Heiligenbrunner, <a href="http://ah9.at/ahsummen.htm">Sum of adjacent numbers (in German)</a>.
%H Nieuw Archief voor Wiskunde 5/6 no. 2, Problems/UWC, Problem C: <a href="http://www.jaapspies.nl/mathfiles/problem20052C.pdf">Solution</a>
%H J. Spies, <a href="http://www.jaapspies.nl/oeis/a111776.sage">Sage program for computing A109814</a>
%F From _Reinhard Zumkeller_, Apr 18 2006: (Start)
%F a(n)*(a(n)+2*A118235(n)1)/2 = n;
%F a(A000079(n)) = 1;
%F a(A000217(n)) = n. (End)
%e Examples provided by _Rainer Rosenthal_, Apr 01 2008:
%e 1 = 1 > a(1) = 1
%e 2 = 2 > a(2) = 1
%e 3 = 1+2 > a(3) = 2
%e 4 = 4 > a(4) = 1
%e 5 = 2+3 > a(5) = 2
%e 6 = 1+2+3 > a(6) = 3
%e a(15) = 5: 15 = 15 (k=1), 15 = 7+8 (k=2), 15 = 4+5+6 (k=3) and 15 = 1+2+3+4+5 (k=5).  _Jaap Spies_, Aug 16 2005
%p A109814:= proc(n) local m, k, d; m := 0; for d from 1 by 2 to n do if n mod d = 0 then k := min(d, 2*n/d): fi; if k > m then m := k fi: od; return(m); end proc; seq(A109814(i),i=1..150); # _Jaap Spies_, Aug 16 2005
%t a[n_] := Reap[Do[If[OddQ[d], Sow[Min[d, 2n/d]]], {d, Divisors[n]}]][[2, 1]] // Max; Table[a[n], {n, 1, 102}]
%o (Sage) sloane.A109814(n) # _Jaap Spies_, Aug 16 2005 (assuming module loaded)
%Y Cf. A001227, A111774, A111775.
%K nonn
%O 1,3
%A _David W. Wilson_
%E Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_
