A338616 a(n) is twice the number of parts in all partitions of n into consecutive parts.

2, 2, 6, 2, 6, 8, 6, 2, 12, 10, 6, 8, 6, 10, 22, 2, 6, 16, 6, 12, 24, 10, 6, 8, 16, 10, 24, 16, 6, 26, 6, 2, 24, 10, 30, 24, 6, 10, 24, 12, 6, 30, 6, 18, 52, 10, 6, 8, 20, 20, 24, 18, 6, 34, 36, 16, 24, 10, 6, 34, 6, 10, 56, 2, 36, 38, 6, 18, 24, 34, 6, 26, 6, 10, 54, 18, 42, 40, 6, 12

Offset: 1

Author

Omar E. Pol, Nov 28 2020

Keywords

• nonn

Comments

a(n) = 6 if and only if n is an odd prime.

a(n) = 2 if and only if n is a power of 2. - Omar E. Pol, Dec 13 2021

Examples

			Illustration of initial terms:
                                        Diagram
n   a(n)                                  _ _
1     2                                 _|1 1|_
2     2                               _|1 _ _ 1|_
3     6                             _|1  |2 2|  1|_
4     2                           _|1   _|   |_   1|_
5     6                         _|1    |2 _ _ 2|    1|_
6     8                       _|1     _| |3 3| |_     1|_
7     6                     _|1      |2  |   |  2|      1|_
8     2                   _|1       _|  _|   |_  |_       1|_
9    12                 _|1        |2  |3 _ _ 3|  2|        1|_
10   10               _|1         _|   | |4 4| |   |_         1|_
11    6             _|1          |2   _| |   | |_   2|          1|_
12    8           _|1           _|   |3  |   |  3|   |_           1|_
13    6         _|1            |2    |  _|   |_  |    2|            1|_
14   10       _|1             _|    _| |4 _ _ 4| |_    |_             1|_
15   22     _|1              |2    |3  | |5 5| |  3|    2|              1|_
16    2    |1                |     |   | |   | |   |     |                1|
...
a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to partitions

Crossrefs

Cf. A054844 (twice the number of partitions of n into consecutive parts), A204217.

Cf. A000079, A001227, A065091, A237591, A237593, A286001, A335616.

Formula

a(n) = 2*A204217(n).