A328365 -id:A328365 - cp's OEIS frontend

A299765 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists the partitions of n into consecutive parts, with the partitions ordered by increasing number of parts.

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

Offset: 1

Omar E. Pol, Feb 26 2018

In the triangle the first partition with m parts appears as the last partition in row A000217(m), m >= 1. - Omar E. Pol, Mar 23 2022

For m >= 0, row 2^m consists of just one element (2^m). - Paolo Xausa, May 24 2025

			Triangle begins:
   [1];
   [2];
   [3], [2, 1];
   [4];
   [5], [3, 2];
   [6], [3, 2, 1];
   [7], [4, 3];
   [8];
   [9], [5, 4], [4, 3, 2];
  [10], [4, 3, 2, 1];
  [11], [6, 5];
  [12], [5, 4, 3];
  [13], [7, 6];
  [14], [5, 4, 3, 2];
  [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17], [9, 8];
  [18], [7, 6, 5], [6, 5, 4, 3];
  [19], [10, 9];
  [20], [6, 5, 4, 3, 2];
  [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1];
  [22], [7, 6, 5, 4];
  [23], [12, 11];
  [24], [9, 8, 7];
  [25], [13, 12], [7, 6, 5, 4, 3];
  [26], [8, 7, 6, 5];
  [27], [14, 13], [10, 9, 8], [7, 6, 5, 4, 3, 2];
  [28], [7, 6, 5, 4, 3, 2, 1];
...
Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
.                                                           _
.                                                         _|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|
...
The diagram is infinite. For more information about the diagram see A286000.
For an amazing connection with sum of divisors function (A000203) see A237593.

Paolo Xausa, Table of n, a(n) for n = 1..10350 (rows 1..500 of triangle, flattened)

Row n has length A204217(n).
Row sums give A245579.
Right border gives A118235.
Column 1 gives A000027.
Records give A000027.
The number of partitions into consecutive parts in row n is A001227(n).
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A328365 (mirror).
Cf. A352425 (a subsequence).
Cf. A000203, A000217, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.

intervals[n_]:=Module[{x,y},SolveValues[(x^2-y^2+x+y)/2==n&&0A299765row[n_]:=Flatten[SortBy[Map[Range[First[#],Last[#],-1]&,intervals[n]],Length]];
nrows=25;Array[A299765row,nrows] (* Paolo Xausa, Jun 19 2022 *)

iscons(p) = my(v = vector(#p-1, k, p[k+1] - p[k])); v == vector(#p-1, i, 1);
row(n) = my(list = List()); forpart(p=n, if (iscons(p), listput(list, Vecrev(p)));); Vec(list); \\ Michel Marcus, May 11 2022

Name clarified by Omar E. Pol, May 11 2022

A245579 Number of odd divisors of n multiplied by n.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 14, 8, 27, 20, 22, 24, 26, 28, 60, 16, 34, 54, 38, 40, 84, 44, 46, 48, 75, 52, 108, 56, 58, 120, 62, 32, 132, 68, 140, 108, 74, 76, 156, 80, 82, 168, 86, 88, 270, 92, 94, 96, 147, 150, 204, 104, 106, 216, 220, 112, 228, 116, 118, 240, 122

Offset: 1

Michael Somos, Jul 26 2014

			G.f. = x + 2*x^2 + 6*x^3 + 4*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 8*x^8 + ...
For n = 10 there are two odd divisors of 10: 1 and 5, so a(10) = 2*10 = 20.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
Omar E. Pol, Comments on A245579.

Cf. A000005, A001227, A001511, A001620, A038040, A285891, A299765, A328362, A328365, A352257, A352505.

seq(n*numtheory:-tau(n/2^padic:-ordp(n,2)), n=1..100); # Robert Israel, Apr 26 2017

a[ n_] := If[ n < 1, 0, n Sum[ Mod[d, 2], {d, Divisors @ n}]];
(* Second program: *)
Table[n DivisorSum[n, 1 &, OddQ], {n, 61}] (* Michael De Vlieger, Apr 24 2017 *)

{a(n) = if( n<1, 0, n * sumdiv(n, d, d%2))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, if( k%2, k * x^k / (1 - x^k)^2), x * O(x^n)), n))};

{a(n) = if( n<1, 0, n * numdiv(n / 2^valuation(n, 2)))} \\ Fast when n has many divisors. Jens Kruse Andersen, Jul 26 2014

from sympy import divisors
def a(n): return n*len(list(filter(lambda i: i%2==1, divisors(n)))) # Indranil Ghosh, Apr 24 2017

from math import prod
from sympy import factorint
def A245579(n): return n*prod(e+1 for e in factorint(n>>(~n&n-1).bit_length()).values()) # Chai Wah Wu, Dec 31 2023

a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = p^e * (e+1) if p>2.
a(n) = n * A001227(n).
G.f.: Sum_{k>0 odd} k * x^k / (1 - x^k)^2.
From Amiram Eldar, Dec 31 2022: (Start)
Dirichlet g.f.: zeta(s-1)^2*(1-1/2^(s-1)).
Sum_{k=1..n} a(k) ~ n^2*log(n)/4 + (4*gamma + 2*log(2) - 1)*n^2/8, where gamma is Euler's constant (A001620). (End)

Edited by N. J. A. Sloane, Apr 27 2022

A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 9, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 5, 5, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Nov 08 2014

Row n has length sigma(n) = A000203(n).
Row sums give n*A000005(n) = A038040(n).
Column 1 is A000027.
Both columns 2 and 3 are A032742, n > 1.
For any k > 0 and t > 0, the sequence contains exactly one run of k consecutive t's. - Rémy Sigrist, Feb 11 2019
From Omar E. Pol, Dec 04 2019: (Start)
The number of parts congruent to 0 (mod m) in row m*n equals sigma(n) = A000203(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The decreasing records in row n give the n-th row of A056538.
Row n has n 1's which are all at the end of the row.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A027750(n,k) parts.
Right border gives A000012. (End)
The r-th row of the triangle begins at index k = A160664(r-1). - Samuel Harkness, Jun 21 2022

			Triangle begins:
   [1];
   [2], [1,1];
   [3], [1,1,1];
   [4], [2,2], [1,1,1,1];
   [5], [1,1,1,1,1];
   [6], [3,3], [2,2,2], [1,1,1,1,1,1];
   [7], [1,1,1,1,1,1,1];
   [8], [4,4], [2,2,2,2], [1,1,1,1,1,1,1,1];
   [9], [3,3,3], [1,1,1,1,1,1,1,1,1];
  [10], [5,5], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1];
  [11], [1,1,1,1,1,1,1,1,1,1,1];
  [12], [6,6], [4,4,4], [3,3,3,3], [2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1];
  [13], [1,1,1,1,1,1,1,1,1,1,1,1,1];
  [14], [7,7], [2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [15], [5,5,5], [3,3,3,3,3], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  [16], [8,8], [4,4,4,4], [2,2,2,2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1];
  ...
For n = 6 the 11 partitions of 6 are [6], [3, 3], [4, 2], [2, 2, 2], [5, 1], [3, 2], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]. There are only four partitions of 6 that contain equal parts so the 6th row of triangle is [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The number of parts equals sigma(6) = A000203(6) = 12. The row sum is A038040(6) = 6*A000005(6) = 6*4 = 24.
From _Omar E. Pol_, Dec 04 2019: (Start)
The structure of the above triangle is as follows:
   1;
   2 11;
   3    111;
   4 22     1111;
   5             11111;
   6 33 222            111111;
   7                          1111111;
   8 44     2222                      11111111;
   9    333                                    111111111;
  ... (End)

Samuel Harkness, Table of n, a(n) for n = 1..10000
Samuel Harkness, Log-log Scatterplot of the first 1000000 terms

Cf. A000005, A000012, A000041, A000203, A000217, A001065, A001227, A002131, A027750, A032742, A038040, A048050, A056538, A126988, A160664, A237593, A245579, A299765, A328365, A309400 (mirror).

A244051row[n_]:=Flatten[Map[ConstantArray[#,n/#]&,Reverse[Divisors[n]]]];
Array[A244051row,10] (* Paolo Xausa, Oct 16 2023 *)

tabf(nn) = {for (n=1, nn, d = Vecrev(divisors(n)); for (i=1, #d, for (j=1, n/d[i], print1(d[i], ", "));); print(););} \\ Michel Marcus, Nov 08 2014

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.

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

A329321 a(n) is the total number of odd parts in all partitions of n into consecutive parts.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 6, 0, 2, 4, 2, 2, 6, 2, 2, 2, 5, 2, 6, 4, 2, 6, 2, 0, 6, 2, 8, 6, 2, 2, 6, 2, 2, 8, 2, 4, 14, 2, 2, 2, 5, 4, 6, 4, 2, 8, 10, 4, 6, 2, 2, 8, 2, 2, 14, 0, 10, 10, 2, 4, 6, 8, 2, 6, 2, 2, 14, 4, 10, 10, 2, 2, 11, 2, 2, 10, 10, 2, 6, 6, 2, 16

Offset: 1

Omar E. Pol, Nov 10 2019

nonn

a(n) = 0 if and only if n is an even power of 2.

			For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are six odd parts, they are [15, 7, 5, 5, 3, 1], so a(15) = 6.

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

Cf. A204217 (total number of parts).

Cf. A000079, A001227, A002131, A237048, A245579, A285898, A285914, A286000, A286001, A299765, A328361, A328365, A329322.

A329321(n) = { my(i=2, t=(n%2)); n--; while(n>0, if(!(n%i), t += (((n/i)%2)+i)\2); n-=i; i++); t }; \\ (After David A. Corneth's program for A204217) - Antti Karttunen, Dec 09 2021

a(n) = A204217(n) - A329322(n).

A329322 a(n) is the total number of even parts in all partitions of n into consecutive parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 3, 5, 1, 1, 4, 1, 4, 6, 3, 1, 2, 3, 3, 6, 4, 1, 7, 1, 1, 6, 3, 7, 6, 1, 3, 6, 4, 1, 7, 1, 5, 12, 3, 1, 2, 5, 6, 6, 5, 1, 9, 8, 4, 6, 3, 1, 9, 1, 3, 14, 1, 8, 9, 1, 5, 6, 9, 1, 7, 1, 3, 13, 5, 11, 10, 1, 4, 10, 3, 1, 9, 8, 3, 6, 6, 1, 18

Offset: 1

Omar E. Pol, Nov 10 2019

nonn

			For n = 15 there are four partitions of 15 into consecutive part, they are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. In total there are five even parts, they are [8, 6, 4, 4, 2], so a(15) = 5.

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

Cf. A204217 (total number of parts).

Cf. A001227, A237048, A245579, A285898, A285914, A286000, A286001, A299765, A328361, A328365, A329321.

A329322(n) = { my(i=2, t=!(n%2)); n--; while(n>0, if(!(n%i), t += (!((n/i)%2)+i)\2); n-=i; i++); t }; \\ (After David A. Corneth's program for A204217) - Antti Karttunen, Dec 09 2021

a(n) = A204217(n) - A329321(n).

A309400 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into equal parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 6, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 5, 5, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11

Offset: 1

Omar E. Pol, Nov 30 2019

The number of parts in row n equals sigma(n) = A000203(n), the sum of the divisors of n. More generally, the number of parts congruent to 0 (mod m) in row m*n equals sigma(n).
The number of parts greater than 1 in row n equals A001065(n), the sum of the aliquot parts of n.
The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.
The number of partitions in row n equals A000005(n), the number of divisors of n.
The number of partitions in row n with an odd number of parts equals A001227(n).
The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).
The sum of row n equals n*A000005(n) = A038040(n).
Records in row n give the n-th row of A027750.
First n rows contain A000217(n) 1's.
The number of k's in row n is A126988(n,k).
The number of odd parts in row n is A002131(n).
The k-th block in row n has A056538(n,k) parts.
Column 1 gives A000012.
Right border gives A000027.

			Triangle begins:
[1];
[1,1], [2];
[1,1,1], [3];
[1,1,1,1], [2,2], [4];
[1,1,1,1,1], [5];
[1,1,1,1,1,1], [2,2,2], [3,3], [6];
[1,1,1,1,1,1,1], [7];
[1,1,1,1,1,1,1,1], [2,2,2,2], [4,4], [8];
[1,1,1,1,1,1,1,1,1], [3,3,3], [9];
[1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10];
[1,1,1,1,1,1,1,1,1,1,1], [11];
[1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3], [4,4,4], [6,6], [12];
[1,1,1,1,1,1,1,1,1,1,1,1,1], [13];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2], [7,7], [14];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [3,3,3,3,3], [5,5,5], [15];
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2,2,2,2], [4,4,4,4], [8,8], [16];
...

Mirror of A244051.

Cf. A000005, A000012, A000027, A000203, A000217, A001065, A001227, A002131, A027750, A038040, A048050, A056538, A126988, A237593, A245579, A299765, A328365.

A328369 Numbers without repeated parts in their partitions into consecutive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 100

Offset: 1

Omar E. Pol, Nov 04 2019

nonn

All primes are terms. - Ivan N. Ianakiev, Nov 05 2019

All powers of 2 are terms. - Omar E. Pol, Nov 19 2019

			The partitions of 9 into consecutive parts are [9], [5, 4], [4, 3, 2]. The 4 is a repeated part, so 9 is not in the sequence.
The partitions of 10 into consecutive parts are [10], [4, 3, 2, 1]. There are no repeated parts, so 10 is in the sequence.

Cf. A001227, A118236 (complement), A286000, A286001, A299765, A328365.

Array[If[Or[PrimeQ@ #, IntegerQ@ Log2@ #], #, # Boole[Count[Tally@ Flatten@ Select[IntegerPartitions[#], Union@ Differences@ # == {-1} &], ?(Last@ # > 1 &)] == 0]] /. 0 -> Nothing &, 60] (* _Michael De Vlieger, Nov 22 2019 *)

cp's OEIS Frontend

A299765 Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists the partitions of n into consecutive parts, with the partitions ordered by increasing number of parts.

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A245579 Number of odd divisors of n multiplied by n.

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A244051 Triangle read by rows in which row n lists the parts of the partitions of n into equal parts, in nonincreasing order.

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

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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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A329321 a(n) is the total number of odd parts in all partitions of n into consecutive parts.

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A329322 a(n) is the total number of even parts in all partitions of n into consecutive parts.

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