A320137 -id:A320137 - cp's OEIS frontend

A319529 Odd numbers that have middle divisors.

1, 9, 15, 25, 35, 45, 49, 63, 77, 81, 91, 99, 117, 121, 135, 143, 153, 165, 169, 187, 195, 209, 221, 225, 231, 247, 255, 273, 285, 289, 299, 315, 323, 325, 345, 357, 361, 375, 391, 399, 405, 425, 435, 437, 441, 459, 475, 483, 493, 513, 525, 527, 529, 551, 561, 567, 575, 589, 609, 621, 625, 627, 651

Offset: 1

Omar E. Pol, Sep 23 2018

nonn

Odd numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
From Felix Fröhlich, Sep 25 2018: (Start)
For the definition of middle divisors, see A067742.
Let t be a term of A005408. Then t is in this sequence iff A067742(t) != 0. (End)
From Hartmut F. W. Hoft, May 24 2022: (Start)
By Theorem 1 (iii) in A067742, the number of middle divisors of a(n) equals the width of the symmetric representation of sigma(a(n)) on the diagonal which equals the triangle entry A249223(n, A003056(n)).
All terms in sequence A016754 have an odd number of middle divisors, forming a subsequence of this sequence; A016754(18) = a(116) = 1225 = 5^2 * 7^2 is the smallest number in A016754 with 3 middle divisors: 25, 35, 49.
Sequence A259417 is a subsequence of this sequence and of A320137 since an even power of a prime has a single middle divisor.
The maximum widths of the center part of the symmetric representation of sigma(a(n)), SRS(a(n)), need not occur at the diagonal. For example, a(304) = 3^3 * 5^3 = 3375, SRS(3375) has 3 parts, its center part has maximum width 3 while its width at the diagonal equals 2 = A067742(3375), and divisors 45 and 75 are the two middle divisors of a(304). (End)

			9 is in the sequence because it's an odd number and the symmetric representation of sigma(9) = 13 has an odd number of parts (more exactly three parts), as shown below:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Intersection of A005408 and A071562.
For more information about the diagram see A237593.
Cf. A000203, A067742, A071090, A236104, A237270, A237271, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319796, A319801, A319802.
Cf. A016754, A237048, A249223, A259417, A320137.

middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, May 24 2022 *)

from itertools import islice, count
from sympy import divisors
def A319529_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(1,startvalue+1-(startvalue&1)),2):
        if any((k <= 2*d**2 < 4*k for d in divisors(k,generator=True))):
            yield k
A319529_list = list(islice(A319529_gen(startvalue=11),40)) # Chai Wah Wu, Jun 09 2022

A320142 Numbers that have exactly two middle divisors.

Original entry on oeis.org

6, 12, 15, 20, 24, 28, 30, 35, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 77, 80, 84, 88, 90, 91, 96, 99, 104, 108, 110, 112, 117, 126, 130, 132, 135, 140, 143, 150, 153, 154, 156, 160, 165, 168, 170, 176, 182, 187, 190, 192, 195, 198, 204, 208, 209, 210, 216, 220, 221, 224, 228, 231, 234, 238, 247, 255, 260

Offset: 1

Omar E. Pol, Oct 06 2018

nonn

Conjecture 1: numbers k with the property that the difference between the number of partitions of k into an odd number of consecutive parts and the number of partitions of k into an even number of consecutive parts is equal to 2.
Conjecture 2: numbers k with the property that symmetric representation of sigma(k) has width 2 on the main diagonal.
By the theorem in A067742 conjecture 2 is true. - Hartmut F. W. Hoft, Aug 18 2024

			15 is in the sequence because 15 has two middle divisors: 3 and 5.
On the other hand, in accordance with the first conjecture, 15 is in the sequence because there are three partitions of 15 into an odd number of consecutive parts: [15], [8, 7], [5, 4, 3, 2, 1], and there is only one partition of 15 into an even number of consecutive parts: [8, 7], therefore the difference of the number of those partitions is 3 - 1 = 2.
On the other hand, in accordance with the second conjecture, 15 is in the sequence because the symmetric representation of sigma(15) = 24 has width 2 on the main diagonal, as shown below in the fourth quadrant:
.                                _
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                               | |
.                          _ _ _|_|
.                      _ _| |      8
.                     |    _|
.                    _|  _|
.                   |_ _|  8
.                   |
.    _ _ _ _ _ _ _ _|
.   |_ _ _ _ _ _ _ _|
.                    8
.

Column 2 of A320051.
First differs from A001284 at a(19).
For the definition of middle divisors see A067742.
Cf. A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320137.

a320142Q[k_] := Length[Select[Divisors[k], k/2<=#^2<2k&]]==2
a320142[n_] := Select[Range[n], a320142Q]
a320142[260] (* Hartmut F. W. Hoft, Aug 20 2024 *)

A320051 Square array read by antidiagonals upwards: T(n,k) is the n-th positive integer with exactly k middle divisors, n >= 1, k >= 0.

Original entry on oeis.org

3, 5, 1, 7, 2, 6, 10, 4, 12, 72, 11, 8, 15, 144, 120, 13, 9, 20, 288, 180, 1800, 14, 16, 24, 400, 240, 3528, 840, 17, 18, 28, 450, 252, 4050, 1080, 3600, 19, 25, 30, 576, 336, 5184, 1260, 7200, 2520, 21, 32, 35, 648, 360, 7056, 1440, 14112, 5040, 28800, 22, 36, 40, 800, 378, 8100, 1680, 14400, 5544

Offset: 1

Omar E. Pol, Oct 04 2018

This is a permutation of the natural numbers.
For the definition of middle divisors see A067742.
Conjecture 1: T(n,k) is also the n-th positive integer j with the property that the difference between the number of partitions of j into an odd number of consecutive parts and the number of partitions of j into an even number of consecutive parts is equal to k.
Conjecture 2: T(n,k) is also the n-th positive integer j with the property that the symmetric representation of sigma(j) has width k on the main diagonal.

			The corner of the square array begins:
   3,  1,  6,  72, 120, 1800,  840,  3600, 2520, 28800, ...
   5,  2, 12, 144, 180, 3528, 1080,  7200, 5040, ...
   7,  4, 15, 288, 240, 4050, 1260, 14112, ...
  10,  8, 20, 400, 252, 5184, 1440, ...
  11,  9, 24, 450, 336, 7056, ...
  13, 16, 28, 576, 360, ...
  14, 18, 30, 648, ...
  17, 25, 35, ...
  19, 32, ...
  21, ...
  ...
In accordance with the conjecture 1, T(1,0) = 3 because there is only one partition of 3 into an odd number of consecutive parts: [3], and there is only one partition of 3 into an even number of consecutive parts: [2, 1], therefore the difference of the number of those partitions is 1 - 1 = 0.
On the other hand, in accordance with the conjecture 2: T(1,0) = 3 because the symmetric representation of sigma(3) = 4 has width 0 on the main diagonal, as shown below:
.    _ _
.   |_ _|_
.       | |
.       |_|
.
In accordance with the conjecture 1, T(1,2) = 6 because there are three partitions of 6 into an odd number of consecutive parts: [6], [3, 2, 1], and there are no partitions of 6 into an even number of consecutive parts, therefore the difference of the number of those partitions is 2 - 0 = 2.
On the other hand, in accordance with the conjecture 2: T(1,2) = 6 because the symmetric representation of sigma(6) = 12 has width 2 on the main diagonal, as shown below:
.    _ _ _ _
.   |_ _ _  |_
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.

Index entries for sequences that are permutations of the natural numbers

Row 1 is A128605.
Column 0 is A071561.
The union of the rest of the columns gives A071562.
Column 1 is A320137.
Column 2 is A320142.
For more information about the diagrams see A237593.
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A067742, A240542, A245092, A249351 (widths), A262626, A279286, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802.

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.

A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

Original entry on oeis.org

2, 8, 18, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 882, 968, 1058, 1250, 1352, 1458, 1682, 1922, 2048, 2178, 2312, 2738, 2888, 3042, 3362, 3698, 3872, 4232, 4418, 4802, 5000, 5202, 5408, 5618, 6050, 6498, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9522

Offset: 1

Hartmut F. W. Hoft, Mar 28 2023

nonn

Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i, j >= 0, k >= 1.

The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.
a(9) = 200 = 2^3 * 5^2 has 10 = 2 * 5 as its single middle divisor, and its symmetric representation of sigma consists of one part of maximum width 2 (A250068), but has width 1 at the diagonal.
a(39) = 6050 = 2^1 * 5^2 * 11^2 has 55 as its single middle divisor; it is the first number in the sequence whose symmetric representation of sigma consists of 3 parts and its central part has maximum width 2, but has width 1 at the diagonal.

Intersection of A001105 and A320137.
Subsequence of A071562 and of A319796.
Cf. A004171, A235791, A237048, A237270, A237271, A237593, A250068.

(* Function a249223[ ] is defined in A320137 *)
a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]
a361905[10000]

a(n) = k when A001105(n) = k and A320137(k) = 1.

A354385 a(n) is the smallest odd number that has n middle divisors.

Original entry on oeis.org

1, 15, 1225, 2145, 99225, 17955, 893025, 51975, 4601025, 315315, 16769025, 855855, 12006225, 2567565, 108056025, 6531525, 385533225, 11486475, 225450225, 16787925, 1329696225, 38513475, 2701400625, 77702625, 6053618025, 80405325, 4846248225, 101846745, 2029052025, 218243025

Offset: 1

Hartmut F. W. Hoft, May 24 2022

nonn

This sequence is nonincreasing since a(5) > a(6), neither is the subsequence a(2n-1), n >= 1, of record odd counts of middle divisors since a(11) = 16769025 > 12006225 = a(13), nor is the subsequence a(2n), n >= 1, of record even counts since a(32) = 413377965 > 334639305 = a(34).
a(21) > 5*10^8.
Further computed values at even indices up to 5*10^8 are a(22, 24, 26, 28, 30, 32, 34) = (38513475, 77702625, 80405325, 101846745, 218243025, 413377965, 334639305).
Observation: At present all known terms >= a(4) are divisible by 3, all >= a(10) are divisible by 7, all >= a(12) are divisible by 11.
Conjecture: For every k, there is an n such that all >= a(n) are divisible by the first k odd primes.

			a(2) = 15 = A319529(3) is the smallest odd number with 2 middle divisors: 3 and 5.
a(3) = 1225 = A319529(116) is the smallest odd number with 3 middle divisors: 25, 35, and 45.

Omar E. Pol, Comments on A354385.

Cf. A067742, A016754, A128605, A235791, A237048, A237593, A245092, A249223, A319529, A320137.

middleDivC[n_] := Length[Select[Divisors[n], Sqrt[n/2]<=#=1&&list[[c]]==0, list[[c]]=k]]; list]
a354385[2*10^7, 20] (* long computation time *)

More terms from Amiram Eldar, Jun 07 2022

Edited by Omar E. Pol at the suggestion of N. J. A. Sloane, Jul 28 2022

A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

Original entry on oeis.org

2, 8, 18, 32, 128, 162, 200, 392, 512, 882, 968, 1352, 1458, 2048, 2178, 3042, 3872, 5000, 5202, 5408, 6498, 8192, 9248, 9522, 11552, 13122, 15138, 16928, 17298, 19208, 26912, 30752, 32768, 36992, 43218, 43808, 46208, 53792, 58482, 59168, 67712, 70688

Offset: 1

Hartmut F. W. Hoft, Aug 29 2023

nonn

Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
(1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
(2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).

			a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
---------------------
  2i+1  /---- p ----/
---------------------
  1       3  ..    3
  3       5  ..   13
  5      11  ..   61
  7      17  ..  251
  9      37  .. 1021
...

Intersection of A361903 and A361905.
Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
The powers of 2 with an odd index (A004171) form a subsequence.
Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.

(* a2[ ] and its support functions are defined in A249223 *)
a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
a365265[{1,75000}]

cp's OEIS Frontend

A319529 Odd numbers that have middle divisors.

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A320142 Numbers that have exactly two middle divisors.

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A320051 Square array read by antidiagonals upwards: T(n,k) is the n-th positive integer with exactly k middle divisors, n >= 1, k >= 0.

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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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A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

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A354385 a(n) is the smallest odd number that has n middle divisors.

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A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

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