%I A109814
%S A109814 1,1,2,1,2,3,2,1,3,4,2,3,2,4,5,1,2,4,2,5,6,4,2,3,5,4,6,7,2,5,2,1,6,4,7,
%T A109814 8,2,4,6,5,2,7,2,8,9,4,2,3,7,5,6,8,2,9,10,7,6,4,2,8,2,4,9,1,10,11,2,8,
               6,
%U A109814 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 A109814 a(n) is the largest k such that n can be written as sum of k consecutive 
               positive integers.
%C A109814 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 (j.spies(AT)hccnet.nl), Aug 16 2005
%D A109814 Nieuw Archief voor Wiskunde 5/6, no. 2, Problems/UWC, Problem C, Jun 
               2005, pp. 181-182
%H A109814 K. S. Brown's Mathpages, <a href="http://www.mathpages.com/home/kmath107.htm">
               Partitions into Consecutive Integers</a>
%H A109814 A. Heiligenbrunner, <a href="http://www.heiligenbrunner.at/main/ahsummen.htm">
               Sum of adjacent numbers (in German)</a>.
%H A109814 Nieuw Archief voor Wiskunde 5/6 no. 2, Problems/UWC, Problem C: <a href="http:/
               /www.jaapspies.nl/mathfiles/problem2005-2C.pdf">Solution</a>
%H A109814 J. Spies, <a href="http://www.jaapspies.nl/oeis/a111776.sage">SAGE program 
               for computing A109814</a>
%F A109814 a(n)*(a(n)+2*A118235(n)-1)/2=n; a(A000079(n))=1; a(A000217(n))=n. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2006
%e A109814 Examples provided by Rainer Rosenthal (r.rosenthal(AT)web.de), Apr 01 
               2008:
%e A109814 1 = 1 ---> a(1) = 1
%e A109814 2 = 2 ---> a(2) = 1
%e A109814 3 = 1+2 ---> a(3) = 2
%e A109814 4 = 4 ---> a(4) = 1
%e A109814 5 = 2+3 ---> a(5) = 2
%e A109814 6 = 1+2+3 ---> a(6) = 3
%e A109814 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 (j.spies(AT)hccnet.nl), Aug 16 2005
%p A109814 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 (j.spies(AT)hccnet.nl), 
               Aug 16 2005
%o A109814 (SAGE) sloane.A109814(n) - Jaap Spies (j.spies(AT)hccnet.nl), Aug 16 
               2005
%Y A109814 Cf. A001227, A111774, A111775.
%Y A109814 Sequence in context: A008342 A002850 A111944 this_sequence A133088 A059982 
               A134388
%Y A109814 Adjacent sequences: A109811 A109812 A109813 this_sequence A109815 A109816 
               A109817
%K A109814 nonn
%O A109814 1,3
%A A109814 David W. Wilson (davidwwilson(AT)comcast.net)
%E A109814 Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 23 2008 at 
               the suggestion of R. J. Mathar