

A109814


a(n) is the largest k such that n can be written as sum of k consecutive positive integers.


16



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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

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


REFERENCES

Nieuw Archief voor Wiskunde 5/6, no. 2, Problems/UWC, Problem C, Jun 2005, pp. 181182.


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Nieuw Archief voor Wiskunde 5/6 no. 2, Problems/UWC, Problem C: Solution
J. Spies, Sage program for computing A109814


FORMULA

From Reinhard Zumkeller, Apr 18 2006: (Start)
a(n)*(a(n)+2*A118235(n)1)/2 = n;
a(A000079(n)) = 1;
a(A000217(n)) = n. (End)


EXAMPLE

Examples provided by Rainer Rosenthal, Apr 01 2008:
1 = 1 > a(1) = 1
2 = 2 > a(2) = 1
3 = 1+2 > a(3) = 2
4 = 4 > a(4) = 1
5 = 2+3 > a(5) = 2
6 = 1+2+3 > a(6) = 3
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


MAPLE

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


MATHEMATICA

a[n_] := Reap[Do[If[OddQ[d], Sow[Min[d, 2n/d]]], {d, Divisors[n]}]][[2, 1]] // Max; Table[a[n], {n, 1, 102}]


PROG

(Sage) sloane.A109814(n) # Jaap Spies, Aug 16 2005 (assuming module loaded)


CROSSREFS

Cf. A001227, A111774, A111775.
Sequence in context: A293909 A002850 A111944 * A133088 A059982 A187801
Adjacent sequences: A109811 A109812 A109813 * A109815 A109816 A109817


KEYWORD

nonn


AUTHOR

David W. Wilson


EXTENSIONS

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar


STATUS

approved



