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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. 181-182.

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

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Last modified October 17 14:20 EDT 2018. Contains 316281 sequences. (Running on oeis4.)