A285900 -id:A285900 - cp's OEIS frontend

A285891 Triangle read by rows: T(n,k) = n*A237048(n,k).

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

Offset: 1

Omar E. Pol, May 02 2017

Conjecture: T(n,k) = n, is also the sum of the parts of the partition of n into k consecutive parts, if such a partition exists, otherwise T(n,k) = 0.

			Triangle begins:
1;
2;
3,   3;
4,   0;
5,   5;
6,   0,  6;
7,   7,  0;
8,   0,  0;
9,   9,  9;
10,  0,  0, 10;
11, 11,  0,  0;
12,  0, 12,  0;
13, 13,  0,  0;
14,  0,  0, 14;
15, 15, 15,  0, 15;
16,  0,  0,  0,  0;
17, 17,  0,  0,  0;
18,  0, 18, 18,  0;
19, 19,  0,  0,  0;
20,  0,  0,  0, 20;
21, 21, 21,  0,  0, 21;
22,  0,  0, 22,  0,  0;
23, 23,  0,  0,  0,  0;
24,  0, 24,  0,  0,  0;
25, 25,  0,  0, 25,  0;
26,  0,  0, 26,  0,  0;
27, 27, 27,  0,  0, 27;
28,  0,  0,  0,  0,  0, 28;
...

Row sums give A245579.
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.
Cf. A196020, A211343, A235791, A236104, A237048, A237591, A237593, A245579, A285900, A285914, A286013.

t(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0); \\ A237048
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(n*t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Oct 20 2019

Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with 1.

Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, 1, 1, (n - 3)/2 zeros, 1.

			Triangle begins:
1;
0, 1;
1, 1, 1;
0, 0, 0, 1;
0, 1, 1, 0, 1;
1, 1, 1, 0, 0, 1;
0, 0, 1, 1, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 1;
0, 1, 1, 2, 1, 0, 0, 0, 1;
1, 1, 1, 1, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1;
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1;
0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 1, 1, 2, 1, 0, 0, 0, 1].

Row sums give A204217.
Column 1 gives A010054, n >= 1.
Leading diagonal gives A000012.
Cf. A000079, A001227, A065091, A066633, A237048, A237593, A245579, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328362.

A285899 Total number of parts in all partitions of all positive integers <= n into consecutive parts.

Original entry on oeis.org

1, 2, 5, 6, 9, 13, 16, 17, 23, 28, 31, 35, 38, 43, 54, 55, 58, 66, 69, 75, 87, 92, 95, 99, 107, 112, 124, 132, 135, 148, 151, 152, 164, 169, 184, 196, 199, 204, 216, 222, 225, 240, 243, 252, 278, 283, 286, 290, 300, 310, 322, 331, 334, 351, 369, 377, 389, 394, 397, 414, 417, 422, 450, 451, 469, 488, 491, 500, 512, 529

Offset: 1

Omar E. Pol, May 02 2017

nonn

Partial sums of A204217.
Sum of first n rows of the triangle A285914.
Where records occur in A328365. - Omar E. Pol, Oct 22 2019
Row sums of A328368. - Omar E. Pol, Nov 04 2019

			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 total number of parts in these four partitions is 11, and a(14) = 43, so a(15) = 43 + 11 = 54.

Cf. A001227, A196020, A204217, A235791, A237048, A237591, A237593, A245092, A285900, A285914, A299765, A328361, A328362, A328365, A328368.

A328362 Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 1, 2, 3, 0, 0, 0, 4, 0, 2, 3, 0, 5, 1, 2, 3, 0, 0, 6, 0, 0, 3, 4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 2, 3, 8, 5, 0, 0, 0, 9, 1, 2, 3, 4, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 11, 0, 0, 3, 4, 5, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 13, 0, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Omar E. Pol, Oct 20 2019

Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with n.

Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, (n - 1)/2, (n + 1)/2, (n - 3)/2 zeros, n.

			Triangle begins:
1;
0, 2;
1, 2, 3;
0, 0, 0, 4;
0, 2, 3, 0, 5;
1, 2, 3, 0, 0, 6;
0, 0, 3, 4, 0, 0, 7;
0, 0, 0, 0, 0, 0, 0, 8;
0, 2, 3, 8, 5, 0, 0, 0, 9;
1, 2, 3, 4, 0, 0, 0, 0, 0, 10;
0, 0, 0, 0, 5, 6, 0, 0, 0,  0, 11;
0, 0, 3, 4, 5, 0, 0, 0, 0,  0,  0, 12;
0, 0, 0, 0, 0, 6, 7, 0, 0,  0,  0,  0, 13;
0, 2, 3, 4, 5, 0, 0, 0, 0,  0,  0,  0,  0, 14;
1, 2, 3, 8,10, 6, 7, 8, 0,  0,  0,  0,  0,  0, 15;
0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16;
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 2, 3, 8, 5, 0, 0, 0, 9].

Row sums give A245579.
Column 1 gives A010054, n => 1.
Leading diagonal gives A000027.
Cf. A000079, A001227, A065091, A138785, A204217, A237048, A237593, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328361.

T(n,k) = k*A328361(n,k).

A328368 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 02 2019

Column k lists k times every nonzero multiple of k in nondecreasing order.

Column k lists the partial sums of the k-th column of triangle A285914.

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

Row sums give A285899.
Row n has length A003056(n).
Column 1 gives A000027.
Column k starts with k in the row A000217(k).
Cf.  A052928, A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.

tt(n, k) = k*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

Original entry on oeis.org

1, 3, 6, 3, 10, 3, 15, 8, 21, 8, 6, 28, 15, 6, 36, 15, 6, 45, 24, 15, 55, 24, 15, 10, 66, 35, 15, 10, 78, 35, 27, 10, 91, 48, 27, 10, 105, 48, 27, 24, 120, 63, 42, 24, 15, 136, 63, 42, 24, 15, 153, 80, 42, 24, 15, 171, 80, 60, 42, 15, 190, 99, 60, 42, 15, 210, 99, 60, 42, 35, 231, 120, 81, 42, 35, 21

Offset: 1

Omar E. Pol, Nov 02 2019

Column k lists the partial sums of the k-th column of triangle A285891.

			Triangle begins:
    1;
    3;
    6,   3;
   10,   3;
   15,   8;
   21,   8,   6;
   28,  15,   6;
   36,  15,   6;
   45,  24,  15;
   55,  24,  15, 10;
   66,  35,  15, 10;
   78,  35,  27, 10;
   91,  48,  27, 10;
  105,  48,  27, 24,
  120,  63,  42, 24, 15;
  136,  63,  42, 24, 15;
  153,  80,  42, 24, 15;
  171,  80,  60, 42, 15;
  190,  99,  60, 42, 15;
  210,  99,  60, 42, 35;
  231, 120,  81, 42, 35, 21;
  253, 120,  81, 64, 35, 21;
  276, 143,  81, 64, 35, 21;
  300, 143, 105, 64, 35, 21;
  325, 168, 105, 64, 60, 21;
  351, 168, 105, 90, 60, 21;
  378, 195, 132, 90, 60, 48;
  406, 195, 132, 90, 60, 48, 28;
...

Row sums give A285900.
Row n has length A003056(n).
Column 1 gives the nonzero terms of A000217.
Column k starts with A000217(k) in the row A000217(k).
Cf. A196020, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285899, A285914, A285891, A286000, A286001, A286013, A299765, A328362, A328365, A328368.

tt(n, k) = n*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

A319895 a(n) is the number of partitions of n into consecutive parts, plus the total number of parts in those partitions.

Original entry on oeis.org

2, 2, 5, 2, 5, 6, 5, 2, 9, 7, 5, 6, 5, 7, 15, 2, 5, 11, 5, 8, 16, 7, 5, 6, 11, 7, 16, 10, 5, 17, 5, 2, 16, 7, 19, 15, 5, 7, 16, 8, 5, 19, 5, 11, 32, 7, 5, 6, 13, 13, 16, 11, 5, 21, 22, 10, 16, 7, 5, 21, 5, 7, 34, 2, 22, 23, 5, 11, 16, 21, 5, 16, 5, 7, 33, 11, 25, 24, 5, 8, 26, 7, 5, 23, 22, 7, 16, 14, 5

Offset: 1

Omar E. Pol, Sep 30 2018

nonn

a(n) is also the total length of all pairs of orthogonal line segments whose horizontal and upper parts are in the n-th row of the diagram associated to partitions into consecutive parts as shown in the Example section.
a(n) = 2 iff n is a power of 2.
a(n) = 5 iff n is an odd prime.

			Illustration of a diagram of partitions into consecutive parts (first 28 rows):
.                                                           _
.                                                         _|1
.                                                       _|2 _
.                                                     _|3  |2
.                                                   _|4   _|1
.                                                 _|5    |3 _
.                                               _|6     _|2|3
.                                             _|7      |4  |2
.                                           _|8       _|3 _|1
.                                         _|9        |5  |4 _
.                                       _|10        _|4  |3|4
.                                     _|11         |6   _|2|3
.                                   _|12          _|5  |5  |2
.                                 _|13           |7    |4 _|1
.                               _|14            _|6   _|3|5 _
.                             _|15             |8    |6  |4|5
.                           _|16              _|7    |5  |3|4
.                         _|17               |9     _|4 _|2|3
.                       _|18                _|8    |7  |6  |2
.                     _|19                 |10     |6  |5 _|1
.                   _|20                  _|9     _|5  |4|6 _
.                 _|21                   |11     |8   _|3|5|6
.               _|22                    _|10     |7  |7  |4|5
.             _|23                     |12      _|6  |6  |3|4
.           _|24                      _|11     |9    |5 _|2|3
.         _|25                       |13       |8   _|4|7  |2
.       _|26                        _|12      _|7  |8  |6 _|1
.     _|27                         |14       |10   |7  |5|7 _
.    |28                           |13       |9    |6  |4|6|7
...
For n = 21 we have that there are four partitions of 21 into consecutive parts, they are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1]. The total number of parts is 1 + 2 + 3 + 6 = 12. Therefore the number of partitions plus the total number of parts is 4 + 12 = 16, so a(21) = 16.
On the other hand, in the above diagram there are four pairs of orthogonal line segments whose horizontal upper part are located on the 21st row, as shown below:
.                   _                     _       _         _
.                  |21                   |11     |8        |6
.                                        |10     |7        |5
.                                                |6        |4
.                                                          |3
.                                                          |2
.                                                          |1
.
The four horizontal line segments have length 1, and the vertical line segments have lengths 1, 2, 3, 6 respectively. Therefore the total length of the line segments is 1 + 1 + 1 + 1 + 1 + 2 + 3 + 6 = 16, so a(21) = 16.

Antti Karttunen, Table of n, a(n) for n = 1..20000

For tables of partitions into consecutive parts see A286000 and A286001.

Cf. A000079, A001227, A065091, A204217, A237048, A237593, A285898, A285899, A285900, A285900, A285901, A285902, A288529, A288772, A288773, A288774, A299765.

A001227(n) = numdiv(n>>valuation(n,2));
A204217(n) = { my(i=2, t=1); n--; while(n>0, t += (i*(n%i==0)); n-=i; i++); t }; \\ From A204217 by David A. Corneth, Apr 28 2017
A319895(n) = (A001227(n)+A204217(n)); \\ Antti Karttunen, Dec 06 2021

a(n) = A001227(n) + A204217(n).

Term a(87) corrected from 6 to 16 by Antti Karttunen, Dec 06 2021

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.

cp's OEIS Frontend

A285891 Triangle read by rows: T(n,k) = n*A237048(n,k).

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A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

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A285899 Total number of parts in all partitions of all positive integers <= n into consecutive parts.

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A328362 Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).

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A328368 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.

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A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

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A319895 a(n) is the number of partitions of n into consecutive parts, plus the total number of parts in those partitions.

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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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