A285901 -id:A285901 - cp's OEIS frontend

A240542 The midpoint of the (rotated) Dyck path from (0, n) to (n, 0) defined by A237593 has coordinates (a(n), a(n)). Also a(n) is the alternating sum of the n-th row of A235791.

1, 2, 2, 3, 3, 5, 5, 6, 7, 7, 7, 9, 9, 9, 11, 12, 12, 13, 13, 15, 15, 15, 15, 17, 18, 18, 18, 20, 20, 22, 22, 23, 23, 23, 25, 26, 26, 26, 26, 28, 28, 30, 30, 30, 32, 32, 32, 34, 35, 36, 36, 36, 36, 38, 38, 40, 40, 40, 40, 42, 42, 42, 44, 45, 45, 47, 47, 47, 47, 49, 49, 52, 52

Offset: 1

Hartmut F. W. Hoft, Apr 07 2014

nonn

The sequence is closely related to the alternating harmonic series.
Its asymptotic behavior is lim_{k -> infinity} a(k)/k = log 2. The relative error is abs(a(k) - k*log(2))/(k*log(2)) <= 2/sqrt(k).
Conjecture 1: the sequence of first positions of the alternating runs of odd and even numbers in a(k) equals A028982. Example: the positions in (1),(2),2,(3),3,5,5,(6),(7),7,7,9,9,9,11,(12),12,(13),13,15,... are 1,2,4,8,9,16,18,... Checked with a Mathematica function through a(1000000).
See A235791, A237591 and A237593 for additional formulas and properties.
Conjecture 2: The sequence of differences a(n) - a(n-1), for n>=1, appears to be equal to A067742(n), the sequence of middle divisors of n; as an empty sum, a(0) = 0, (which was conjectured by Michel Marcus in the entry A237593). Checked with the two respective Mathematica functions up to n=100000. - Hartmut F. W. Hoft, Jul 17 2014
The number of occurrences of n is A259179(n). - Omar E. Pol, Dec 11 2016
Conjecture 3: a(n) is also the difference between the total number of partitions of all positive integers <= n into an odd number of consecutive parts, and the total number of partitions of all positive integers <= n into an even number of consecutive parts. - Omar E. Pol, Apr 28 2017
Conjecture 4: a(n) is also the total number of central subparts of all symmetric representations of sigma of all positive integers <= n. - Omar E. Pol, Apr 29 2017
a(n) is also the sum of the odd-indexed terms of the n-th row of the triangle A237591. - Omar E. Pol, Jun 20 2018
a(n) is the total number of middle divisors of all positive integers <= n (after Michel Marcus's conjecture in A237593). - Omar E. Pol, Aug 18 2021

			From _Omar E. Pol_, Dec 22 2020: (Start)
Illustration of initial terms in two ways in accordance with the sum of the odd-indexed terms of the rows of A237591:
.
n   a(n)                               _              _
1    1                               _|_|            |_|_
2    2                             _|_ _|            |_ _|
3    2                           _|_ _|              |_ _|_
4    3                         _|_ _ _|              |_ _ _|
5    3                       _|_ _ _|  _             |_ _ _|_ _
6    5                     _|_ _ _ _| |_|            |_ _ _ _|_|
7    5                   _|_ _ _ _|   |_|            |_ _ _ _|_|_
8    6                 _|_ _ _ _ _|  _|_|            |_ _ _ _ _|_|_
9    7               _|_ _ _ _ _|   |_ _|            |_ _ _ _ _|_ _|
10   7             _|_ _ _ _ _ _|   |_|              |_ _ _ _ _ _|_|
11   7           _|_ _ _ _ _ _|    _|_|              |_ _ _ _ _ _|_|_ _
12   9         _|_ _ _ _ _ _ _|   |_ _|              |_ _ _ _ _ _ _|_ _|
13   9       _|_ _ _ _ _ _ _|     |_ _|              |_ _ _ _ _ _ _|_ _|
14   9     _|_ _ _ _ _ _ _ _|    _|_|  _             |_ _ _ _ _ _ _ _|_|_ _
15  11   _|_ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _|_ _|_|_
16  12  |_ _ _ _ _ _ _ _ _|     |_ _| |_|            |_ _ _ _ _ _ _ _ _|_ _|_|
...
                  Figure 1.                                  Figure 2.
.
Figure 1 shows the illustration of initial terms taken from the isosceles triangle of A237593. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Figure 2 shows the illustration of initial terms taken from an octant of the pyramid described in A244050 and A245092. For n = 16 there are (9 + 2 + 1) = 12 cells in the 16th row of the diagram, so a(16) = 12.
Note that if we fold each level (or row) of that isosceles triangle of A237593 we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n).
(End).

Cf. A028982, A067742, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A071562, A259176, A259179, A279387, A286001, A322141, A338204, A338758.

a[n_] := Sum[(-1)^(k + 1) Ceiling[(n + 1)/k - (k + 1)/2], {k, 1, Floor[-1/2 + 1/2 Sqrt[8 n + 1]]}]; Table[a[n], {n, 40}]

a(n) = sum(k=1, floor(-1/2 + 1/2*sqrt(8*n + 1)), (-1)^(k + 1)*ceil((n + 1)/k - (k + 1)/2)); \\ Indranil Ghosh, Apr 21 2017

from sympy import sqrt
import math
def a(n): return sum((-1)**(k + 1) * int(math.ceil((n + 1)/k - (k + 1)/2)) for k in range(1, int(math.floor(-1/2 + 1/2*sqrt(8*n + 1))) + 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 21 2017

a(n) = Sum_{k = 1..A003056(n)} (-1)^(k+1) A235791(n,k).
a(n) = A285901(n) - A285902(n), assuming the conjecture 3. - Omar E. Pol, Feb 15 2018
a(n) = n - A322141(n). - Omar E. Pol, Dec 22 2020

More terms from Omar E. Pol, Apr 16 2014

Definition edited by N. J. A. Sloane, Dec 20 2020

A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 1, 1, 4, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 1, 3, 2, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 3, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4

Offset: 1

Naohiro Nomoto, May 15 2003

Number of ways to write n as the sum of an odd number of consecutive integers. - Vladeta Jovovic, Aug 28 2007
Number of odd divisors of n less than sqrt(2*n). - Vladeta Jovovic, Sep 16 2007
Conjecture: a(n) is also the number of subparts in an octant of the symmetric representation of sigma(n). - Omar E. Pol, Feb 22 2017

			For n=6: 6 has two ways -- (d=3; 3|6), 1+2+3=6; and (d=1; 1|6), 6=6 -- so a(6)=2.

Peter Kagey, Table of n, a(n) for n = 1..10000
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
William Lowell Putnam Competition, Function A(k) in Problem B6, 2015.

Cf. A001227, A054843, A082637, A082662, A237593, A285901.

N:= 1000: # to get a(1) to a(N)
g:= add(x^(k*(2*k-1))/(1-x^(2*k-1)), k=1..floor(sqrt(N/2))):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Dec 08 2015

a(n) = my(q = sqrt(2*n)); sumdiv(n, d, (d%2) && (d < q)); \\ Michel Marcus, Jul 04 2014

G.f.: Sum_{k>0} x^(k*(2*k-1))/(1-x^(2*k-1)). - Vladeta Jovovic, Aug 25 2004
Conjecture: a(n) = A067742(n) + A131576(n). - Omar E. Pol, Feb 22 2017
Conjecture: a(n) = A001227(n) - A131576(n). - Omar E. Pol, Apr 18 2017

A285902 Total number of partitions of all positive integers <= n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 14, 15, 16, 16, 17, 18, 20, 20, 21, 22, 23, 23, 25, 26, 27, 28, 29, 30, 32, 32, 33, 34, 35, 36, 38, 39, 40, 40, 41, 42, 44, 45, 46, 47, 49, 49, 51, 52, 53, 54, 55, 56, 58, 58, 60, 61, 62, 63, 65, 66, 67, 67, 68, 69, 72, 73, 74, 76, 77, 77, 79, 80, 81, 82

Offset: 1

Omar E. Pol, May 02 2017

nonn

a(n) is also the total number of odd divisors of k greater than sqrt(2*k), for k = 1..n.
Conjecture: a(n) is also the total number of pairs of equidistant subparts of all symmetric representations of sigma of all positive integers <= n.
For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Partial sums of A131576.

Cf. A001227, A060831, A082647, A196020, A235791, A236104, A237048, A237270, A237591, A237593, A237270, A279387, A245092, A262626, A285901.

a(n) = A060831(n) - A285901(n).

A317292 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 0.

Original entry on oeis.org

0, 4, 8, 14, 20, 26, 36, 42, 50, 60, 70, 76, 92, 98, 108, 124, 136, 142, 160, 166, 182, 198, 208, 214, 238, 250, 260, 276, 294, 300

Offset: 0

Omar E. Pol, Jul 27 2018

All terms are even numbers.

Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.

			Illustration of initial terms (n = 1..9):
.                                                       _ _ _ _
.                                         _ _ _        |_ _ _  |_
.                             _ _ _      |_ _ _|       |_ _ _| |_|_
.                   _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.           _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.     _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.     4      8        14         20           26             36
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _| |_|_ _      |_ _ _| |_|_ _  |     |_ _ _| |_|_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           42                  50                     60
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 300 edges, 239 vertices and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | | |_ _  |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _  | |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _| |_  |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_  |_  |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_  |_ _  | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_  | |_ _  | | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_  |_|_  | | |_|_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_  | | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_  | |_|_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  | |_  |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_  |_  | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_  |_|_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A317293 (number of vertices).
Cf. A060831 (number of regions or subparts).
Compare with A317109 (analog for the diagram that contains only parts).
First differs from A317109 at a(6).
Cf. A000203, A001227, A196020, A235791, A237048, A237590, A237591, A237270, A237271, A237593, A245092, A244050, A262626, A280850, A280851, A280940, A285901, A294723, A296508.

a(n) = A317293(n) + A060831(n) - 1 (Euler's formula).

A317293 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 1.

Original entry on oeis.org

1, 4, 7, 11, 16, 20, 28, 32, 39, 46, 54, 58, 72, 76, 84, 96, 107, 111, 126, 130, 144, 156, 164, 168, 190, 199, 207, 219, 235, 239

Offset: 0

Omar E. Pol, Jul 27 2018

Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _| |_|_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             28
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _| |_|_ _      |_ _ _| |_|_ _  |     |_ _ _| |_|_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           32                  39                     46
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 239 vertices, 300 edges and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | | |_ _  |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _  | |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _| |_  |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_  |_  |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_  |_ _  | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_  | |_ _  | | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_  |_|_  | | |_|_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_  | | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_  | |_|_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  | |_  |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_  |_  | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_  |_|_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A317292 (number of edges).
Cf. A060831 (number of regions or subparts).
Compare with A294723 (analog for the diagram that contains only parts).
First differs from A294723 at a(6).
Cf. A000203, A196020, A235791, A237048, A237590, A237591, A237270, A237271, A237593, A245092, A244050, A262626, A280850, A280851, A280940, A285901, A296508, A317109.

a(n) = A317292(n) - A060831(n) + 1 (Euler's formula).

A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 2, 1, 7, 8, 9, 4, 3, 2, 10, 11, 12, 5, 4, 3, 13, 14, 15, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 18, 7, 6, 5, 19, 20, 6, 5, 4, 3, 2, 21, 8, 7, 6, 22, 23, 24, 9, 8, 7, 25, 7, 6, 5, 4, 3, 26, 27, 10, 9, 8, 28, 7, 6, 5, 4, 3, 2, 1, 29, 30, 11, 10, 9, 8, 7, 6, 5, 4

Offset: 1

Omar E. Pol, Mar 15 2022

Conjecture: the total number of parts in all partitions of n into an odd number of consecutive parts equals the sum of odd divisors of n that are <= A003056(n). In other words: row n has A341309(n) terms.

The first partition with 2*m - 1 parts appears in the row A000384(m), m >= 1.

			Triangle begins:
   [1];
   [2];
   [3],
   [4];
   [5];
   [6], [3, 2, 1];
   [7];
   [8];
   [9], [4, 3, 2];
  [10];
  [11];
  [12], [5, 4, 3];
  [13];
  [14];
  [15], [6, 5, 4], [5, 4, 3, 2, 1];
  [16];
  [17];
  [18], [7, 6, 5];
  [19];
  [20], [6, 5, 4, 3, 2];
  [21], [8, 7, 6];
  [22];
  [23];
  [24], [9, 8, 7];
  [25], [7, 6, 5, 4, 3];
  [26];
  [27], [10, 9, 8];
  [28], [7, 6, 5, 4, 3, 2, 1];
  ...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions 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], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
                                                           _
                                                         _|1|
                                                       _|2  |
                                                     _|3    |
                                                   _|4      |
                                                 _|5       _|
                                               _|6        |3|
                                             _|7          |2|
                                           _|8           _|1|
                                         _|9            |4  |
                                       _|10             |3  |
                                     _|11              _|2  |
                                   _|12               |5    |
                                 _|13                 |4    |
                               _|14                  _|3   _|
                             _|15                   |6    |5|
                           _|16                     |5    |4|
                         _|17                      _|4    |3|
                       _|18                       |7      |2|
                     _|19                         |6     _|1|
                   _|20                          _|5    |6  |
                 _|21                           |8      |5  |
               _|22                             |7      |4  |
             _|23                              _|6      |3  |
           _|24                               |9       _|2  |
         _|25                                 |8      |7    |
       _|26                                  _|7      |6    |
     _|27                                   |10       |5   _|
    |28                                     |9        |4  |7|
...
The diagram is infinite.
For more information about the diagram see A286000.

Subsequence of A299765.
Row sums give A352257.
Column 1 gives A000027.
Records give A000027.
Row n contains A082647(n) of the mentioned partitions.
Cf. A000384, A003056, A067742, A204217, A237048, A237591, A237593, A240542, A245092, A285574, A285901, A286000, A286001, A320051, A320137, A320142, A341309, A351824.

A353149 Sum of the odd-indexed terms in the n-th row of the triangle A196020.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 13, 15, 20, 19, 21, 28, 25, 27, 37, 31, 33, 44, 37, 42, 52, 43, 45, 60, 54, 51, 68, 56, 57, 83, 61, 63, 84, 67, 81, 92, 73, 75, 100, 90, 81, 113, 85, 87, 130, 91, 93, 124, 104, 114, 132, 103, 105, 143, 126, 120, 148, 115, 117, 175, 121, 123, 180, 127, 150, 173, 133, 135, 180, 175

Offset: 1

Omar E. Pol, Apr 26 2022

nonn

a(n) is the total number of steps in all odd-indexed double-staircases of the diagram of A196020 with n levels (see the example).
a(n) is also the total number of steps in all odd-indexed double-staircases of the diagram described in A335616 with n levels that have at least one step in the bottom level of the diagram.
Sigma(n) <= a(n).
The graph of the sum-of-divisors function A000203 is intermediate between the graph of this sequence and the graph of A353154 (see link). - Omar E. Pol, May 13 2022

			For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the odd-indexed terms is 29 + 7 + 1 = 37, so a(15) = 37.
Illustration of a(15) = 37:
Level                                   Diagram
.                                          _
1                                        _|1|_
2                                      _|1   1|_
3                                    _|1       1|_
4                                  _|1           1|_
5                                _|1       _       1|_
6                              _|1        |1|        1|_
7                            _|1          | |          1|_
8                          _|1           _| |_           1|_
9                        _|1            |1   1|            1|_
10                     _|1              |     |              1|_
11                   _|1               _|     |_               1|_
12                 _|1                |1       1|                1|_
13               _|1                  |         |                  1|_
14             _|1                   _|    _    |_                   1|_
15            |1                    |1    |1|    1|                    1|
.
The diagram has 37 steps, so a(15) = 37.

David A. Corneth, Table of n, a(n) for n = 1..10000
OEIS Plot 2, A353149 vs A000203

Cf. A000203, A057640, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353154.

a(n) = { my(r = A196020row(n)); sum(i = 0, (#r-1)\2, r[2*i + 1]) }
A196020row(n) = { my(res, qc); qc = (sqrtint(8*n + 1) - 1)\2; res = vector(qc); for(i = 1, qc, cn = n - binomial(i + 1, 2); if(cn % i == 0, res[i] = 2*(cn/i) + 1 ) ); res } \\ David A. Corneth, Apr 28 2022

a(n) = A000203(n) + A353154(n).

a(n) = A209246(n) - A353154(n).

A353154 Sum of the even-indexed terms in the n-th row of the triangle A196020.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 5, 0, 7, 1, 9, 0, 11, 3, 13, 0, 15, 5, 17, 0, 20, 7, 21, 0, 23, 9, 28, 0, 27, 11, 29, 0, 36, 13, 33, 1, 35, 15, 44, 0, 39, 17, 41, 3, 52, 19, 45, 0, 47, 21, 60, 5, 51, 23, 54, 0, 68, 25, 57, 7, 59, 27, 76, 0, 66, 29, 65, 9, 84, 31, 69, 0, 71, 33, 97, 11, 75, 36, 77, 0

Offset: 1

Omar E. Pol, Apr 27 2022

nonn

Conjecture: indices of zeros give A082662.
a(n) is the total number of steps in all even-indexed double-staircases of the diagram of A196020 with n levels.
a(n) is also the total number of steps in all even-indexed double-staircases of the diagram described in A335616 with n levels that have at least one step in the bottom level of the diagram.
The graph of the sum-of-divisors function A000203 is intermediate between the graph of A353149 and the graph of this sequence (see the Links section). - Omar E. Pol, May 13 2022

			For n = 15 the 15th row of the triangle A196020 is [29, 13, 7, 0, 1]. The sum of the even-indexed terms is 13 + 0 = 13, so a(15) = 13.

OEIS Plot 2, A353154 vs A000203

Cf. A000203, A057641, A082662, A196020, A209246, A211343, A235791, A236104, A237591, A237593, A285901, A335616, A347186, A353149.

a(n) = A353149(n) - A000203(n).

a(n) = A209246(n) - A353149(n).

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

A285999 Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 10, 12, 12, 14, 16, 18, 18, 20, 22, 24, 24, 28, 30, 32, 32, 34, 36, 40, 40, 42, 44, 46, 46, 50, 52, 54, 56, 58, 60, 64, 64, 66, 68, 70, 72, 76, 78, 80, 80, 82, 84, 88, 90, 92, 94, 98, 98, 102, 104, 106, 108, 110, 112, 116, 116, 120, 122, 124, 126, 130, 132, 134, 134, 136, 138, 144, 146, 148, 152

Offset: 1

Omar E. Pol, May 14 2017

nonn

Conjecture 1: a(n) is also twice the total number of partitions of all positive integers <= n into an even number of consecutive parts.
Conjecture 2: a(n) is also the total number of equidistant subparts of the symmetric representations of sigma of all positive integers <= n. Thus a(n) is also the total number of equidistant subparts in the terraces of the stepped pyramid with n levels described in A245092.
For more information about the "subparts" of the symmetric representation of sigma see A279387 and A237593.

Partial sums of A281009.

Cf. A060831, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A279387, A285901, A285902.

Accumulate@ Table[DivisorSum[n, 1 &, OddQ] - DivisorSum[n, 1 &, Sqrt[n/2] <= # < Sqrt[2 n] &], {n, 78}] (* Michael De Vlieger, May 18 2017 *)

Conjecture: a(n) = A060831(n) - A240542(n).

Conjecture: a(n) = 2*A285902(n).

cp's OEIS Frontend

A240542 The midpoint of the (rotated) Dyck path from (0, n) to (n, 0) defined by A237593 has coordinates (a(n), a(n)). Also a(n) is the alternating sum of the n-th row of A235791.

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A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

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A285902 Total number of partitions of all positive integers <= n into an even number of consecutive parts.

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A317292 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 0.

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A317293 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 1.

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A352425 Irregular triangle read by rows in which row n lists the partitions of n into an odd number of consecutive parts.

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A353149 Sum of the odd-indexed terms in the n-th row of the triangle A196020.

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A353154 Sum of the even-indexed terms in the n-th row of the triangle A196020.

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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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