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Search: id:A109814
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| A109814 |
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a(n) is the largest k such that n can be written as sum of k consecutive positive integers. |
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+0
8
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| 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; listen)
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OFFSET
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1,3
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COMMENT
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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
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REFERENCES
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Nieuw Archief voor Wiskunde 5/6, no. 2, Problems/UWC, Problem C, Jun 2005, pp. 181-182
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LINKS
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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
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FORMULA
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a(n)*(a(n)+2*A118235(n)-1)/2=n; a(A000079(n))=1; a(A000217(n))=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 18 2006
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EXAMPLE
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Examples provided by Rainer Rosenthal (r.rosenthal(AT)web.de), 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 (j.spies(AT)hccnet.nl), Aug 16 2005
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MAPLE
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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
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PROGRAM
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(SAGE) sloane.A109814(n) - Jaap Spies (j.spies(AT)hccnet.nl), Aug 16 2005
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CROSSREFS
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Cf. A001227, A111774, A111775.
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KEYWORD
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nonn,new
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
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Edited by njas, Aug 23 2008 at the suggestion of R. J. Mathar
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