A054844 Number of ways to write n as the sum of any number of consecutive integers (including the trivial one-term sum n = n).

2, 2, 4, 2, 4, 4, 4, 2, 6, 4, 4, 4, 4, 4, 8, 2, 4, 6, 4, 4, 8, 4, 4, 4, 6, 4, 8, 4, 4, 8, 4, 2, 8, 4, 8, 6, 4, 4, 8, 4, 4, 8, 4, 4, 12, 4, 4, 4, 6, 6, 8, 4, 4, 8, 8, 4, 8, 4, 4, 8, 4, 4, 12, 2, 8, 8, 4, 4, 8, 8, 4, 6, 4, 4, 12, 4, 8, 8, 4, 4, 10, 4, 4, 8, 8, 4, 8, 4, 4, 12, 8, 4, 8, 4, 8, 4, 4, 6, 12, 6

Offset: 1

Author

Henry Bottomley, Apr 13 2000

Keywords

• easy
• nonn

Comments

a(n) = twice the number of odd divisors of n. That is, if d is the divisor function and q is the exponent of the largest power of 2 dividing n, then the a(n) equals 2*d(n)/(q+1). - Andrew Niedermaier, Jul 20 2003

Moebius transform is period 2 sequence [2, 0, ...]. - Michael Somos, Sep 20 2005

a(n) is twice the number of partitions of n into consecutive parts. - Omar E. Pol, Nov 28 2020

Examples

			a(3) = 4 because 3 = (-2)+(-1)+0+1+2+3 or 0+1+2 or 1+2 or 3; a(13) = 4 because 13 = (-12)+...+13 or (-5)+...+7 or 6+7 or 13.
From _Omar E. Pol_, Nov 28 2020: (Start)
Illustration of initial terms:
                                        Diagram
n   a(n)                                  _ _
1     2                                 _|1 1|_
2     2                               _|1 _ _ 1|_
3     4                             _|1  |1 1|  1|_
4     2                           _|1   _|   |_   1|_
5     4                         _|1    |1 _ _ 1|    1|_
6     4                       _|1     _| |1 1| |_     1|_
7     4                     _|1      |1  |   |  1|      1|_
8     2                   _|1       _|  _|   |_  |_       1|_
9     6                 _|1        |1  |1 _ _ 1|  1|        1|_
10    4               _|1         _|   | |1 1| |   |_         1|_
11    4             _|1          |1   _| |   | |_   1|          1|_
12    4           _|1           _|   |1  |   |  1|   |_           1|_
13    4         _|1            |1    |  _|   |_  |    1|            1|_
14    4       _|1             _|    _| |1 _ _ 1| |_    |_             1|_
15    8     _|1              |1    |1  | |1 1| |  1|    1|              1|_
16    2    |1                |     |   | |   | |   |     |                1|
...
a(n) is the number of horizontal toothpicks in the n-th level of the diagram. (End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A001227, A010054, A054843, A237593, A335616.

Programs

PARI

a(n)=2*sumdiv(n,d,d%2)

PARI

A054844(n) = (2*numdiv(n>>valuation(n, 2))); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = 2*A001227(n). - Andrew Niedermaier, Jul 20 2003

G.f.: Sum_{k>0} 2x^k/(1-x^(2k)) = Sum_{k>0} 2x^(2k-1)/(1-x^(2k-1)). - Michael Somos, Sep 20 2005

a(n) = A010054(n) + A335616(n). - Omar E. Pol, Nov 28 2020

Extensions

Corrected and extended by Michael Somos, Apr 26 2000