A109814 a(n) is the largest k such that n can be written as sum of k consecutive positive integers.

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

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

The numbers that can be written as a sum of k consecutive positive integers are those in column k of A141419 (as a triangle). - Peter Munn, Mar 01 2019

The numbers that cannot be written as a sum of two or more consecutive positive integers are the powers of 2. So a(n) = 1 iff n = 2^k for k >= 0. - Bernard Schott, Mar 03 2019

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).

Jaap Spies, Problem C NAW 5/6 nr. 2 June 2005, July 2005 (solution to problem below).

Jaap Spies, Sage program for computing A109814

Universitaire Wiskunde Competitie, Problem C, Nieuw Archief voor Wiskunde, 5/6, no. 2, Problems/UWC, Jun 2005, pp. 181-182.

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) for n in range(1, 20)]
# Jaap Spies, Aug 16 2005
(Python)
from sympy import divisors
def a(n): return max(min(d, 2*n//d) for d in divisors(n) if d&1)
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 23 2022

Crossrefs

Cf. A001227, A111774, A111775, A141419, A138591.

Cf. A000079 (powers of 2), A000217 (triangular numbers).

Sequence in context: A002850 A355248 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