A286014 -id:A286014 - cp's OEIS frontend

A286013 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the positive integers starting with k, interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

1, 2, 3, 2, 4, 0, 5, 3, 6, 0, 3, 7, 4, 0, 8, 0, 0, 9, 5, 4, 10, 0, 0, 4, 11, 6, 0, 0, 12, 0, 5, 0, 13, 7, 0, 0, 14, 0, 0, 5, 15, 8, 6, 0, 5, 16, 0, 0, 0, 0, 17, 9, 0, 0, 0, 18, 0, 7, 6, 0, 19, 10, 0, 0, 0, 20, 0, 0, 0, 6, 21, 11, 8, 0, 0, 6, 22, 0, 0, 7, 0, 0, 23, 12, 0, 0, 0, 0, 24, 0, 9, 0, 0, 0, 25, 13, 0, 0, 7, 0

Offset: 1

Omar E. Pol, Apr 30 2017

Conjecture 1: T(n,k) is the largest part of the partition of n into k consecutive parts, if T(n,k) > 0.
Conjecture 2: row sums give A286015.
Trapezoidal interpretation from Peter Munn, Jun 18 2017: (Start)
There is one to one correspondence between nonzero T(n,k) and trapezoidal area patterns of n dots on a triangular grid, if we include the limiting cases of triangular patterns, straight lines (k=1) or a single dot (k=n=1). The corresponding pattern has T(n,k) dots in its longest side, k dots in the two adjacent sides and T(n,k)-k+1 dots in the fourth side (where a count of 1 dot may be understood as signifying that side's absence).
Reason: From the definition, for k >= 1, m >= 0, T(A000217(k)+km,k) = k+m, where A000217(k) = k(k+1)/2, the k-th triangular number. First element of column k is T(A000217(k),k) = k: this matches a triangular pattern of A000217(k) dots with 3 sides of k dots. Looking at this pattern as k rows of 1..k dots, extend each row by m dots to create a trapezoidal pattern of A000217(k)+km dots with a longest side of k+m dots and adjacent sides of k dots: this matches T(A000217(k)+km,k) = k+m. As nonzero elements in column k occur at intervals of k, every nonzero T(n,k) has a match. Every trapezoidal pattern can be produced by extending a triangular pattern as described, so they all have a match.
The truth of conjecture 1 follows, since each nonzero T(n,k) = k+m corresponds to a trapezoidal pattern of n dots having k rows with lengths (1+m)..(k+m).
The A270877 sieve is related to this sequence because it eliminates n if it is the sum of consecutive numbers whose largest term has survived the sifting (which may likewise be seen in terms of a trapezoidal dot pattern and its longest side). So the sieve eliminates n if any lesser numbers in A270877 are in row n of this sequence.
(End)

			Triangle begins:
1;
2;
3,   2;
4,   0;
5,   3;
6,   0,  3;
7,   4,  0;
8,   0,  0;
9,   5,  4;
10,  0,  0,  4;
11,  6,  0,  0;
12,  0,  5,  0;
13,  7,  0,  0;
14,  0,  0,  5;
15,  8,  6,  0,  5;
16,  0,  0,  0,  0;
17,  9,  0,  0,  0;
18,  0,  7,  6,  0;
19, 10,  0,  0,  0;
20,  0,  0,  0,  6;
21, 11,  8,  0,  0,  6;
22,  0,  0,  7,  0,  0;
23, 12,  0,  0,  0,  0;
24,  0,  9,  0,  0,  0;
25, 13,  0,  0,  7,  0;
26,  0,  0,  8,  0,  0;
27, 14, 10,  0,  0,  7;
28,  0,  0,  0,  0,  0,  7;
...
In accordance with the conjecture, for n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The largest parts are 15, 8, 6, 5, respectively, so the 15th row of the triangle is [15, 8, 6, 0, 5].

Michael De Vlieger, Table of n, a(n) for n = 1..10944 (Rows 1 <= n <= 528, 528 being first row with 32 columns).

Row n has length A003056(n).
Column k starts in row A000217(k).
The number of positive terms in row n is A001227(n), the number of partitions of n into consecutive parts.
The last positive term in row n is in column A109814(n).
Cf. A196020, A204217, A211343, A235791, A237048, A237591, A237593, A245579, A270877, A286014, A286015.

With[{n = 7}, DeleteCases[#, m_ /; m < 0] & /@ Transpose@ Table[Apply[Join @@ {ConstantArray[-1, #2 - 1], Array[(k + #/k) Boole[Mod[#, k] == 0] &, #1 - #2 + 1, 0]} &, # (# + 1)/2 & /@ {n, k}], {k, n}]] // Flatten (* Michael De Vlieger, Jul 21 2017 *)

For k >= 1, m >= 0, T(A000217(k)+km,k) = k+m. - Peter Munn, Jun 19 2017

A286015 Sum of largest parts of all partitions of n into consecutive parts.

Original entry on oeis.org

1, 2, 5, 4, 8, 9, 11, 8, 18, 14, 17, 17, 20, 19, 34, 16, 26, 31, 29, 26, 46, 29, 35, 33, 45, 34, 58, 35, 44, 58, 47, 32, 70, 44, 70, 57, 56, 49, 82, 50, 62, 78, 65, 53, 114, 59, 71, 65, 84, 76, 106, 62, 80, 98, 106, 67, 118, 74, 89, 106, 92, 79, 153, 64, 124

Offset: 1

Omar E. Pol, Apr 30 2017

nonn

If n is a power of 2 then a(n) = n, the same as A286014(n).

Conjecture: this is also the row sums of A286013.

			For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. The sum of the largest parts is 15 + 8 + 6 + 5 = 34, so a(15) = 34.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000079, A006128, A046746, A204217, A211343, A245579, A286013, A286014.

Table[Total[Select[IntegerPartitions@ n, Or[Length@ # == 1, Union@ Differences@ # == {-1}] &][[All, 1]]], {n, 65}] (* Michael De Vlieger, Jul 21 2017 *)

More terms from Alois P. Heinz, May 01 2017

A379630 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 5, 2, 5, 1, 6, 1, 3, 1, 7, 3, 7, 1, 8, 1, 9, 4, 2, 3, 9, 1, 10, 1, 5, 1, 11, 5, 11, 1, 12, 3, 3, 1, 13, 6, 13, 1, 14, 2, 7, 1, 15, 7, 4, 1, 5, 3, 15, 1, 16, 1, 17, 8, 17, 1, 18, 5, 3, 9, 3, 1, 19, 9, 19, 1, 20, 2, 5, 1, 21, 10, 6, 1, 7, 3, 21, 1, 22, 4, 11, 1, 23, 11, 23, 1, 24, 7, 3, 1

Offset: 1

Omar E. Pol, Dec 28 2024

Theorem of correspondence between the partitions of n into k consecutive parts and the odd divisors of n: given a partition of n into k consecutive parts if k is odd then the corresponding odd divisor of n is k, otherwise if k is even then the corresponding odd divisor of n is the sum of any pair of conjugate parts of the partition (for example the sum of the largest part and the smallest part).
Conjecture: the first A001227(n) terms in the n-th row are also the absolute values of the n-th row of A341971.
The last A001227(n) terms in the n-th row are also the mirror of the n-th row of A261697.

			Triangle begins:
   1,  1;
   2,  1;
   3,  1,  3,  1;
   4,  1;
   5,  2,  5,  1;
   6,  1,  3,  1;
   7,  3,  7,  1;
   8,  1;
   9,  4,  2,  3,  9,  1;
  10,  1,  5,  1;
  11,  5, 11,  1;
  12,  3,  3,  1;
  13,  6, 13,  1;
  14,  2,  7,  1;
  15,  7,  4,  1,  5,  3, 15,  1;
  16,  1;
  17,  8, 17,  1;
  18,  5,  3,  9,  3,  1;
  19,  9, 19,  1;
  20,  2,  5,  1;
  21, 10,  6,  1,  7,  3, 21,  1;
  ...
For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1].
On the other hand the odd divisors of 21 are [1, 3, 7, 21].
To determine how these partitions are related to the odd divisors we follow the two rules of the theorem as shown below:
The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1.
The second partition is [11, 10] and the number of parts is 2 and 2 even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21.
The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3.
The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7.
Summarizing in a table:
  --------------------------------------
              Correspondence
  --------------------------------------
    Partitions of 21              Odd
    into consecutive           divisors
         parts                   of 21
  -------------------         ----------
   [21]   ....................     1
   [11, 10]   ................    21
   [8, 7, 6]  ................     3
   [6, 5, 4, 3, 2, 1]  .......     7
.
Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the smallest parts at the top as shown below:
  ------------------------------------------
           Conjugate correspondence
  ------------------------------------------
    Partitions of 21              Odd
    into consecutive           divisors
    parts as columns             of 21
  -------------------     ------------------
   21   10    6    1       7    3   21    1
    |   11    7    2       |    |    |    |
    |    |    8    3       |    |    |    |
    |    |    |    4       |    |    |    |
    |    |    |    5       |    |    |    |
    |    |    |    6       |    |    |    |
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for smallest parts and odd divisors as shown below:
  -------------------     ------------------
    Smallest parts           Odd divisors
  -------------------     ------------------
   21   10    6    1       7    3   21    1
    |    |    |    |_______|    |    |    |
    |    |    |_________________|    |    |
    |    |___________________________|    |
    |_____________________________________|
.
So the 21st row of the triangle is [21, 10, 6, 1, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem:
.                                          _ _
                                         _|1|1|_
                                       _|2 _|_ 1|_
                                     _|3  |1|3|  1|_
                                   _|4   _| | |_   1|_
                                 _|5    |2 _|_ 5|    1|_
                               _|6     _| |1|3| |_     1|_
                             _|7      |3  | | |  7|      1|_
                           _|8       _|  _| | |_  |_       1|_
                         _|9        |4  |2 _|_ 3|  9|        1|_
                       _|10        _|   | |1|5| |   |_         1|_
                     _|11         |5   _| | | | |_  11|          1|_
                   _|12          _|   |3  | | |  3|   |_           1|_
                 _|13           |6    |  _| | |_  |   13|            1|_
               _|14            _|    _| |2 _|_ 7| |_    |_             1|_
             _|15             |7    |4  | |1|5| |  3|   15|              1|_
           _|16              _|     |   | | | | |   |     |_               1|_
         _|17               |8     _|  _| | | | |_  |_    17|                1|_
       _|18                _|     |5  |3  | | |  9|  3|     |_                 1|_
     _|19                 |9      |   |  _| | |_  |   |     19|                  1|_
   _|20                  _|      _|   | |2 _|_ 5| |   |_      |_                   1|_
  |21                   |10     |6    | | |1|7| | |    3|     21|                    1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangle A286001.
Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092.
.

Column 1 gives A000027.
Right border gives A000012.
The sum of row n equals A286014(n) + A000593(n).
The length of row n is A054844(n) = 2*A001227(n).
The partitions of n into consecutive parts are in the n-th row of A299765. See also A286000 and A286001.
The odd divisors of n are in the n-th row of A182469. See also A261697 and A261699.
Cf. A196020, A204217, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A341971.

A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 6, 1, 7, 3, 8, 9, 4, 2, 10, 1, 11, 5, 12, 3, 13, 6, 14, 2, 15, 7, 4, 1, 16, 17, 8, 18, 5, 3, 19, 9, 20, 2, 21, 10, 6, 1, 22, 4, 23, 11, 24, 7, 25, 12, 3, 26, 5, 27, 13, 8, 2, 28, 1, 29, 14, 30, 9, 6, 4, 31, 15, 32, 33, 16, 10, 3, 34, 7, 35, 17, 5, 2, 36, 11, 1, 37, 18, 38, 8, 39, 19, 12, 4, 40, 6, 41, 20

Offset: 1

Omar E. Pol, Dec 31 2024

Row n gives the first A001227(n) terms of the n-th row of A379630.

			Triangle begins:
   1;
   2;
   3,  1;
   4;
   5,  2;
   6,  1;
   7,  3;
   8;
   9,  4,  2;
  10,  1;
  11,  5;
  12,  3;
  13,  6;
  14,  2;
  15,  7,  4,  1;
  16;
  17,  8;
  18,  5,  3;
  19,  9;
  20,  2;
  21, 10,  6,  1;
  22,  4;
  23, 11;
  24,  7;
  25, 12,  3;
  26,  5;
  27, 13,  8,  2;
  28,  1;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |1|
                                                 _|4   _| |
                                               _|5    |2 _|
                                             _|6     _| |1|
                                           _|7      |3  | |
                                         _|8       _|  _| |
                                       _|9        |4  |2 _|
                                     _|10        _|   | |1|
                                   _|11         |5   _| | |
                                 _|12          _|   |3  | |
                               _|13           |6    |  _| |
                             _|14            _|    _| |2 _|
                           _|15             |7    |4  | |1|
                         _|16              _|     |   | | |
                       _|17               |8     _|  _| | |
                     _|18                _|     |5  |3  | |
                   _|19                 |9      |   |  _| |
                 _|20                  _|      _|   | |2 _|
               _|21                   |10     |6   _| | |1|
             _|22                    _|       |   |4  | | |
           _|23                     |11      _|   |   | | |
         _|24                      _|       |7    |  _| | |
       _|25                       |12       |    _| |3  | |
     _|26                        _|        _|   |5  |  _| |
   _|27                         |13       |8    |   | |2 _|
  |28                           |         |     |   | | |1|
  ...
The diagram is also the left part of the diagram of A379630.
The geometrical structure is the same as the diagram of A237591.

Positive terms of A211343.
Absolute values of A341971.
Column 1 gives A000027.
Right border gives A118235.
Row lengths give A001227.
Row sums give A286014.
Subsequence of A286001 and of A299765 and of A379630.
For the largest parts see A379633.
Cf. A196020, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379631, A379634.

cp's OEIS Frontend

A286013 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the positive integers starting with k, interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

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A286015 Sum of largest parts of all partitions of n into consecutive parts.

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A379630 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.

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A379632 Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

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