A285902 -id:A285902 - 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

A285901 Total number of partitions of all positive integers <= n into an odd number of consecutive parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 20, 21, 22, 24, 25, 27, 29, 30, 31, 33, 35, 36, 38, 40, 41, 44, 45, 46, 48, 49, 52, 54, 55, 56, 58, 60, 61, 64, 65, 66, 70, 71, 72, 74, 76, 78, 80, 81, 82, 85, 87, 89, 91, 92, 93, 96, 97, 98, 102, 103, 105, 108, 109, 110, 112, 115, 116, 119, 120, 121, 124, 125, 128, 130

Offset: 1

Omar E. Pol, May 02 2017

nonn

a(n) is also the total number of odd divisors of k less than sqrt(2*k), for k = 1..n.
Conjecture: a(n) is also the total number of subparts present (totally or partially) in an octant of the 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 A082647.

Cf. A001227, A060831, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A244250, A262618, A279387, A285902.

a(n) = A060831(n) - A285902(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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A285901 Total number of partitions of all positive integers <= n into an odd number of 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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A285999 Total number of odd divisors of all positive integers <= n, minus the total number of middle divisors of all positive integers <= n.

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