A204217 -id:A204217 - cp's OEIS frontend

A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

1, 4, 1, 6, 2, 1, 12, 6, 2, 1, 10, 4, 3, 2, 1, 24, 10, 8, 3, 2, 1, 14, 12, 5, 4, 3, 2, 1, 32, 14, 12, 10, 4, 3, 2, 1, 27, 8, 7, 6, 5, 4, 3, 2, 1, 40, 27, 16, 14, 12, 5, 4, 3, 2, 1, 22, 20, 18, 8, 7, 6, 5, 4, 3, 2, 1, 72, 22, 20, 18, 16, 14, 6, 5, 4, 3, 2, 1, 26, 24, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, May 05 2020

			Array begins:
     k  0   1   2   3   4   5   6   7   8   9  10
   n +------------------------------------------------
   1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...
   5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...
   6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...
   7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...
   8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...
   9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...
  10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
...

Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).
Every diagonal starting with 1 gives A000027.
Sequences of number of parts related to column k: A000203 (k=0), A204217 (k=1), A066839 (k=2) (conjectured), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6).
Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).
Polygonal  numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).
Cf. A245579, A323345, A334466.

nmax = 13;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
T[n_, k_] := n col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

T(n,k) = n*A323345(n,k).

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

Original entry on oeis.org

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

Omar E. Pol, Nov 28 2020

nonn

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

			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.

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

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

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

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

A352446 Total number of parts in all partitions of n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 8, 4, 2, 0, 2, 4, 8, 0, 2, 4, 2, 0, 8, 4, 2, 8, 2, 4, 8, 0, 2, 4, 2, 8, 8, 4, 2, 0, 2, 4, 8, 8, 2, 4, 12, 0, 8, 4, 2, 8, 2, 4, 8, 0, 12, 4, 2, 8, 8, 4, 2, 0, 2, 4, 18, 8, 2, 16, 2, 0, 8, 4, 2, 8, 12, 4, 8, 0, 2, 16, 2

Offset: 1

Omar E. Pol, Mar 16 2022

nonn

			For n = 21  the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1]. The total number of parts in these two partitions is equal to 2 + 6 = 8, so a(21) = 8.
On the other hand consider the diagram below which is formed by the even-indexed staircase walks from the diagram of A286000.
The diagram is infinite and we have that:
The m-th staircase walk starts at row A014105(m).
The number of horizontal line segment in the n-th row equals A131576(n), the number of partitions of n into an even number of consecutive parts.
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.
---------------------------------------------
   n    a(n)              Diagram
---------------------------------------------
   1      0
   2      0                               _
   3      2                              |2
   4      0                             _|1
   5      2                            |3
   6      0                           _|2
   7      2                          |4
   8      0                         _|3
   9      2                        |5     _
  10      4                       _|4    |4
  11      2                      |6      |3
  12      0                     _|5      |2
  13      2                    |7       _|1
  14      4                   _|6      |5
  15      2                  |8        |4
  16      0                 _|7        |3
  17      2                |9         _|2
  18      4               _|8        |6
  19      2              |10         |5
  20      0             _|9          |4   _
  21      8            |11          _|3  |6
  22      4           _|10         |7    |5
  23      2          |12           |6    |4
  24      0         _|11           |5    |3
  25      2        |13            _|4    |2
  26      4       _|12           |8     _|1
  27      8      |14             |7    |7
  28      0      |13             |6    |6
...
For n = 21 the number of horizontal line segment in the 21th row of the diagram equals A131576(21) = 2, the number of partitions of 21 into an even number of consecutive parts.
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 21-th level of the diagram is equal to 2 + 6 = 8, so a(21) = 8.

For more information about the diagram see A286000 and A237593.

Cf. A014105, A131576, A204217, A299765, A341309, A352505.

a(n) = A204217(n) - A341309(n), conjectured.

Conjecture: G.f.: Sum_{k>=1} 2*k*x^(k*(2*k+1))/(1-x^(2*k)). - Vaclav Kotesovec, Oct 23 2024

A352733 Least number k such that the total number of parts in all partitions of k into consecutive parts is a record.

Original entry on oeis.org

1, 3, 6, 9, 15, 21, 30, 35, 45, 63, 90, 105, 135, 165, 189, 210, 225, 297, 315, 495, 525, 630, 825, 945, 1155, 1365, 1485, 1575, 2205, 2475, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6930, 8085, 8190, 8775, 9009, 9450, 10395, 12285, 15015, 16065, 17325, 20475

Offset: 1

Omar E. Pol, Mar 30 2022

nonn

Where records occur in A204217.

Cf. A001227, A237048, A286000, A299765, A352734.

More terms from Amiram Eldar, Apr 01 2022

A352734 Records of the total number of parts in all partitions of a positive integer into consecutive parts.

Original entry on oeis.org

1, 3, 4, 6, 11, 12, 13, 15, 26, 28, 34, 48, 51, 53, 60, 67, 69, 72, 111, 132, 139, 160, 163, 245, 257, 269, 291, 338, 375, 406, 477, 660, 707, 717, 775, 780, 800, 851, 872, 887, 930, 933, 984, 1023, 1406, 1492, 1644, 1664, 1984, 2128, 2211, 2358, 2391, 2444, 2796, 3015, 4020, 4410, 4510, 4735, 4793, 4976, 5161, 5232, 5312, 5454, 5784

Offset: 1

Omar E. Pol, Mar 30 2022

nonn

Records in A204217.

Cf. A001227, A237048, A299765, A286000, A352733.

a(n) = A204217(A352733(n)).

More terms from Max Alekseyev, Oct 12 2023

A284475 Total number of parts in all partitions of n into equal parts, minus the total number of parts in all partitions of n into consecutive parts.

Original entry on oeis.org

0, 2, 1, 6, 3, 8, 5, 14, 7, 13, 9, 24, 11, 19, 13, 30, 15, 31, 17, 36, 20, 31, 21, 56, 23, 37, 28, 48, 27, 59, 29, 62, 36, 49, 33, 79, 35, 55, 44, 84, 39, 81, 41, 75, 52, 67, 45, 120, 47, 83, 60, 89, 51, 103, 54, 112, 68, 85, 57, 151, 59, 91, 76, 126, 66, 125, 65, 117, 84, 127, 69, 182, 71, 109, 97, 131, 75, 148

Offset: 1

Omar E. Pol, May 03 2017

nonn

Observation: at least for the first 78 terms of this sequence the values of n where a(n) = n - 2 coincide with the odd numbers of A082664.

			For n = 10 the partitions of 10 into equal parts are [10], [5, 5], [2, 2, 2, 2, 2] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. The total number of parts is 18. On the other hand, the partitions of 10 into consecutive parts are [10] and [4, 3, 2, 1]. The total number of parts is 5, so a(10) = 18 - 5 = 13.

Cf. A000203, A000918, A082647, A082664, A204217, A285914.

Conjecture: a(n) = A000203(n) - A204217(n).

a(2^k) = A000918(k+1), k>=0.

A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 5, 5, 6, 5, 6, 7, 7, 6, 8, 7, 6, 9, 9, 9, 10, 9, 9, 10, 11, 11, 9, 10, 12, 11, 12, 10, 13, 13, 12, 10, 14, 13, 12, 14, 15, 15, 15, 14, 15, 16, 15, 15, 14, 15, 17, 17, 15, 14, 15, 18, 17, 18, 18, 15, 19, 19, 18, 18, 15, 20, 19, 18, 18, 20, 21, 21, 21, 18, 20, 21, 22, 21, 21, 22, 20, 21

Offset: 1

Omar E. Pol, Nov 09 2019

T(n,k) is also the positive integer whose partition into k consecutive parts is associated to the k-th vertex, from left to right, of the largest Dyck path of the symmetric representation of sigma(n). For more information see A237593.

Also this triangle can be constructed replacing every zero of triangle A285891 with the previous positive integer from the same column.

			Triangle begins:
   1;
   2;
   3,  3;
   4,  3;
   5,  5;
   6,  5,  6;
   7,  7,  6;
   8,  7,  6;
   9,  9,  9;
  10,  9,  9, 10;
  11, 11,  9, 10;
  12, 11, 12, 10;
  13, 13, 12, 10;
  14, 13, 12, 14;
  15, 15, 15, 14, 15;
  16, 15, 15, 14, 15;
  17, 17, 15, 14, 15;
  18, 17, 18, 18, 15;
  19, 19, 18, 18, 15;
  20, 19, 18, 18, 20;
  21, 21, 21, 18, 20, 21;
  22, 21, 21, 22, 20, 21;
  23, 23, 21, 22, 20, 21;
  24, 23, 24, 22, 20, 21;
  25, 25, 24, 22, 25, 21;
  26, 25, 24, 26, 25, 21;
  27, 27, 27, 26, 25, 27;
  28, 27, 27, 26, 25, 27, 28;
...

Column k stars with A000217(k) in the row A000217(k).
Row n has length A003056(n).
Cf. A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

A347286 a(n) is n minus the number of odd divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 11, 15, 15, 15, 17, 18, 17, 20, 21, 22, 22, 24, 23, 26, 27, 26, 29, 31, 29, 32, 31, 33, 35, 36, 35, 38, 39, 38, 41, 42, 39, 44, 45, 46, 46, 47, 47, 50, 51, 50, 51, 54, 53, 56, 57, 56, 59, 60, 57, 63, 61, 62, 65, 66, 65, 66

Offset: 1

Omar E. Pol, Sep 12 2021

a(n) is n minus the number of partitions of n into consecutive parts.
This definition is in accordance with the diagram as shown below in the Example section which also appears in many sequences related to A237048, A237591, A237593 and possible others.
a(n) is also the number of zeros in the n-th row of A285898.

			For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18. There are three odd divisors: 1, 3, 9, so a(18) = 18 - 3 = 15.
On the other hand the partitions of 18 into consecutive parts are [18], [7, 6, 5], [6, 5, 4, 3]. There are three of such partitions, so a(18) = 18 - 3 = 15.
Illustration of initial terms:
.
   n   a(n)                   Diagram                 _
   1     0                                          _|x|
   2     1                                        _|x _|
   3     1                                      _|x  |x|
   4     3                                    _|x   _| |
   5     3                                  _|x    |x _|
   6     4                                _|x     _| |x|
   7     5                              _|x      |x  | |
   8     7                            _|x       _|  _| |
   9     6                          _|x        |x  |x _|
  10     8                        _|x         _|   | |x|
  11     9                      _|x          |x   _| | |
  12    10                    _|x           _|   |x  | |
  13    11                  _|x            |x    |  _| |
  14    12                _|x             _|    _| |x _|
  15    11              _|x              |x    |x  | |x|
  16    15            _|x               _|     |   | | |
  17    15          _|x                |x     _|  _| | |
  18    15        _|x                 _|     |x  |x  | |
  19    17      _|x                  |x      |   |  _| |
  20    18    _|x                   _|      _|   | |x _|
  21    17   |x                    |x      |x    | | |x|
...
In the above diagram the number of x's in row n equals A001227(n), the number of partitions n into consecutive parts.
a(n) is the number of square cells in row n that do not contain a "x".
In other words: a(n) is the number of square cells in row n that do not have a horizontal line segment above.

Cf. A001227, A196020, A204217, A235791, A237048, A237591, A237593, A238005, A245092, A285898, A286000, A286001.

a[n_] := n - DivisorSigma[0, n/2^IntegerExponent[n, 2]]; Array[a, 70] (* Amiram Eldar, Sep 12 2021 *)

a(n) = n - sumdiv(n, d, d%2); \\ Michel Marcus, Sep 12 2021

a(n) = n - A001227(n).

cp's OEIS Frontend

A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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

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A352446 Total number of parts in all partitions of n into an even number of consecutive parts.

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A352733 Least number k such that the total number of parts in all partitions of k into consecutive parts is a record.

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A352734 Records of the total number of parts in all partitions of a positive integer into consecutive parts.

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A284475 Total number of parts in all partitions of n into equal parts, minus the total number of parts in all partitions of n into consecutive parts.

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A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.

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A347286 a(n) is n minus the number of odd divisors of n.

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