cp's OEIS Frontend

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

%I A338616 #37 Dec 18 2021 22:20:57
%S A338616 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,
%T A338616 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,
%U A338616 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
%N A338616 a(n) is twice the number of parts in all partitions of n into consecutive parts.
%C A338616 a(n) = 6 if and only if n is an odd prime.
%C A338616 a(n) = 2 if and only if n is a power of 2. - _Omar E. Pol_, Dec 13 2021
%H A338616 Antti Karttunen, <a href="/A338616/b338616.txt">Table of n, a(n) for n = 1..20000</a>
%H A338616 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A338616 a(n) = 2*A204217(n).
%e A338616 Illustration of initial terms:
%e A338616                                         Diagram
%e A338616 n   a(n)                                  _ _
%e A338616 1     2                                 _|1 1|_
%e A338616 2     2                               _|1 _ _ 1|_
%e A338616 3     6                             _|1  |2 2|  1|_
%e A338616 4     2                           _|1   _|   |_   1|_
%e A338616 5     6                         _|1    |2 _ _ 2|    1|_
%e A338616 6     8                       _|1     _| |3 3| |_     1|_
%e A338616 7     6                     _|1      |2  |   |  2|      1|_
%e A338616 8     2                   _|1       _|  _|   |_  |_       1|_
%e A338616 9    12                 _|1        |2  |3 _ _ 3|  2|        1|_
%e A338616 10   10               _|1         _|   | |4 4| |   |_         1|_
%e A338616 11    6             _|1          |2   _| |   | |_   2|          1|_
%e A338616 12    8           _|1           _|   |3  |   |  3|   |_           1|_
%e A338616 13    6         _|1            |2    |  _|   |_  |    2|            1|_
%e A338616 14   10       _|1             _|    _| |4 _ _ 4| |_    |_             1|_
%e A338616 15   22     _|1              |2    |3  | |5 5| |  3|    2|              1|_
%e A338616 16    2    |1                |     |   | |   | |   |     |                1|
%e A338616 ...
%e A338616 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.
%Y A338616 Cf. A054844 (twice the number of partitions of n into consecutive parts), A204217.
%Y A338616 Cf. A000079, A001227, A065091, A237591, A237593, A286001, A335616.
%K A338616 nonn
%O A338616 1,1
%A A338616 _Omar E. Pol_, Nov 28 2020