A299777 -id:A299777 - cp's OEIS frontend

A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

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

Offset: 1

Omar E. Pol, Jun 08 2018

The middle divisors of n are the divisors in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins (rows 1..16):
1;
1;
0;
2;
0;
2, 3;
0;
2;
3;
0;
0;
3, 4;
0;
0;
3, 5;
4;
...
For n = 6 the middle divisors of 6 are 2 and 3, so row 6 is [2, 3].
For n = 7 there are no middle divisors of 7, so row 7 is [0].
For n = 8 the middle divisor of 8 is 2, so row 8 is [2].
For n = 72 the middle divisors of 72 are 6, 8 and 9, so row 72 is [6, 8, 9].

Michael De Vlieger, Table of n, a(n) for n = 1..14002 (rows 1 <= n <= 10^4)

Row sums give A071090.
The number of nonzero terms in row n is A067742(n).
Nonzero terms give A303297.
Indices of the rows where there are zeros give A071561.
Indices of the rows where there are nonzero terms give A071562.
Cf. A027750, A281007, A299777.

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> {0}, {n, 80}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

row(n) = my(v=select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); if (#v, v, [0]); \\ Michel Marcus, Aug 04 2022

A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 3, 4, 5, 4, 6, 5, 4, 7, 5, 6, 4, 5, 7, 6, 5, 8, 6, 7, 5, 9, 6, 8, 7, 5, 6, 9, 7, 8, 6, 10, 7, 9, 8, 6, 11, 7, 10, 6, 8, 9, 7, 11, 8, 10, 9, 7, 12, 8, 11, 9, 10, 7, 13, 8, 12, 7, 9, 11, 10, 8, 13, 9, 12, 10, 11, 8, 14, 9, 13, 8, 10, 12, 15, 11, 9, 14, 8, 10, 13, 11, 12, 9, 15

Offset: 1

Omar E. Pol, Apr 30 2018

The middle divisors of k (see A299761) are the divisors in the half-open interval [sqrt(k/2), sqrt(k*2)), k >= 1.

			The middle divisor of 1 is 1, so a(1) = 1.
The middle divisor of 2 is 1, so a(2) = 1.
There are no middle divisors of 3.
The middle divisor of 4 is 2, so a(3) = 2.
There are no middle divisors of 5.
The middle divisors of 6 are 2 and 3, so a(4) = 2 and a(5) = 3.
There are no middle divisors of 7.
The middle divisor of 8 is 2, so a(6) = 2.
The middle divisor of 9 is 3, so a(7) = 3.
There are no middle divisors of 10.
There are no middle divisors of 11.
The middle divisors of 12 are 3 and 4, so a(8) = 3 and a(9) = 4.

Michel Marcus, Table of n, a(n) for n = 1..6934
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16.
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 345, labeling n, and showing composite prime powers in gold, squarefree composites in green, numbers neither squarefree nor composite in blue, and highlighting products of composite prime powers in large light blue.
Michael De Vlieger, Plot (n,d) at (x,y) for middle divisors d | n and n <= 2^16, with same color function as above so as to show patterns according to prime power decomposition of n.

Concatenate the nonzero rows of A299761 (that is, the nonzero terms of A299761).

Cf. A027750, A067742, A071090, A071562, A281007, A299777.

Table[Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] /. {} -> Nothing, {n, 135}] // Flatten (* Michael De Vlieger, Jun 14 2018 *)

lista(nn) = {my(list = List()); for (n=1, nn, my(v = select(x->((x >= sqrt(n/2)) && (x < sqrt(n*2))), divisors(n))); for (i=1, #v, listput(list, v[i]));); Vec(list);} \\ Michel Marcus, Mar 26 2023

A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 50, 54, 56, 60, 64, 66, 70, 72, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 120, 126, 128, 130, 132, 140, 144, 150, 154, 156, 160, 162, 168, 170, 176, 180, 182, 190, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 238, 240, 242, 252, 256

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
An even number A005843 is in this sequence iff A067742(t) != 0.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
From Hartmut F. W. Hoft, Mar 28 2023: (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)), SRS(a(n)), on the diagonal which equals the triangle entry A249223(n, A003056(n)). The maximum widths of the center part of SRS(a(n)) need not occur at the diagonal.
For example, a(7) = 2 * 3^2 = 18, SRS(18) has a single part with maximum width 2 while its width at the diagonal equals 1 = A067742(18), and divisor 3 is the only middle divisor of a(7). (End)

			6 is in the sequence because it's an even number and the symmetric representation of sigma(6) = 12 has an odd number of parts (more exactly only one part), as shown below:
.    _ _ _ _
.   |_ _ _  |_ 12
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.
Also 50 is in the sequence because it's an even number and the symmetric representation of sigma(50) = 93 has an odd number of parts (more exactly three parts), they are [39, 15, 39].
a(34) = 110 = 2 * 5 * 11 has 10 and 11 as its middle divisors, and SRS(a(34)) has 3 parts and width 2 at the diagonal. -  _Hartmut F. W. Hoft_, Mar 28 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071562.

Cf. A000203, A067742, A071090, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319801, A319802.

filter:= n -> ormap(t -> t^2 >= n/2 and t^2 < 2*n, numtheory:-divisors(n)):
select(filter, 2*[$1..1000]); # Robert Israel, Mar 29 2023

middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#Hartmut F. W. Hoft, Mar 28 2023 *)

Name clarified by Omar E. Pol, Mar 28 2023

A319529 Odd numbers that have middle divisors.

Original entry on oeis.org

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

A319802 Even numbers without middle divisors.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 230, 232, 236, 244, 246, 248, 250, 254, 258, 262, 268, 274, 278, 282, 284

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Even numbers k such that the symmetric representation of sigma(k) has an even number of parts.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
Let p be a prime > 5. Then a(n) is a number of the form m*p where m is an even number < sqrt(p). - David A. Corneth, Sep 28 2018
First differs from A244894 at a(51) = 230. - R. J. Mathar, Oct 04 2018
Is this twice A101550? - Omar E. Pol, Oct 04 2018
This sequence is not twice A101550: first differs at a(57) = 250 != 254 = 2*A101550(57). - Michael S. Branicky, Oct 14 2021

			10 is in the sequence because it's an even number and the symmetric representation of sigma(10) = 18 has an even number of parts as shown below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A071561.
Cf. A244894 (a subsequence).
Cf. A000203, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319801.

from sympy import divisors
def ok(n):
    if n < 2 or n%2 == 1: return False
    return not any(n//2 <= d*d < 2*n for d in divisors(n, generator=True))
print(list(filter(ok, range(285)))) # Michael S. Branicky, Oct 14 2021

A319801 Odd numbers without middle divisors.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 79, 83, 85, 87, 89, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145, 147, 149, 151, 155, 157, 159, 161, 163, 167, 171, 173, 175, 177

Offset: 1

Omar E. Pol, Sep 28 2018

nonn

Odd numbers k such that the symmetric representation of sigma(k) has an even number of parts.
All odd primes (A065091) are in the sequence.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.

			21 is in the sequence because it's an odd number and the symmetric representation of sigma(21) = 32 has an even number of parts (more exactly four parts), as shown below:
.     _ _ _ _ _ _ _ _ _ _ _ 11
.    |_ _ _ _ _ _ _ _ _ _ _|
.                          |
.                          |
.                          |_ _ _ 5
.                          |_ _  |_
.                              |_ _|_
.                                  | |_ 5
.                                  |_  |
.                                    | |
.                                    |_|_ _ _ _ 11
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            | |
.                                            |_|
.

Intersection of A005408 and A071561.

Cf. A000203, A065091, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239931, A239933, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319802.

A320137 Numbers that have only one middle divisor.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 81, 98, 100, 121, 128, 162, 169, 196, 200, 225, 242, 256, 289, 324, 338, 361, 392, 441, 484, 512, 529, 578, 625, 676, 722, 729, 784, 841, 882, 961, 968, 1024, 1058, 1089, 1156, 1250, 1352, 1369, 1444, 1458, 1521, 1681, 1682, 1849, 1922, 1936, 2025, 2048, 2116

Offset: 1

Omar E. Pol, Oct 06 2018

nonn

Conjecture 1: sequence consists of 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 1.
Conjecture 2: sequence consists of numbers k with the property that the symmetric representation of sigma(k) has width 1 on the main diagonal.
Conjecture 3: all powers of 2 are in the sequence.
From Hartmut F. W. Hoft, May 24 2022: (Start)
Every number in this sequence is a square or twice a square, i.e., this sequence is a subsequence of A028982, and conjectures 2 and 3 are true (see the link for proofs). Furthermore, all odd numbers in this sequence are squares and form subsequences of A016754 and of A319529.
Every number k in this sequence has the form k = 2^m * q^2, m >= 0, q >= 1 odd, where for any divisor e of q^2 smaller than the largest divisor of q^2 that is less than or equal to row(q^2) = floor((sqrt(8*q^2 + 1) - 1)/2) the inequalities 2^(m+1) * e < row(n) hold (see the link for a proof).
The smallest odd square not in this sequence is 1225 = 35^2 = (5*7)^2 since it has the 3 middle divisors 25, 35, 49 and the width of the symmetric representation of sigma(1225) at the diagonal equals 3. However, the squares of odd primes in this sequence are a subsequence of A259417.
The smallest even square not in this sequence is 144 = 12^2 = (2*2*3)^2 since it has the 3 middle divisors 9, 12, 16 and the width of the symmetric representation of sigma(144) at the diagonal equals 3.
The smallest twice square not in this sequence is 72 = 2 * (2*3)^2 = 2^3 * 3^2 since it has the 3 middle divisors 6, 8, 9 and the width of the symmetric representation of sigma(72) at the diagonal equals 3.
Apart from the powers of 2 in the infinite first row, the numbers in the sequence can be arranged as an irregular triangle with each row containing the finitely many numbers q^2, 2 * q^2, 4 * q^2, ..., 2^m * q^2 satisfying the condition stated above, as shown below:
    1     2     4     8     16     32     64     128     256 ...
    9    18    36
   25    50   100   200
   49    98   196   392    784
   81   162   324
  121   242   484   968   1936   3872
  169   338   676  1352   2704   5408  10816
  225
  289   578  1156  2312   4624   9248  18496   36992
  361   722  1444  2888   5776  11552  23104   46208
  441   882
  529  1058  2116  4232   8464  16928  33856   67712  135424
  625  1250  2500  5000
  729  1458  2916
  841  1682  3364  6728  13456  26912  53824  107648  215296
  ...
(End)

			9 is in the sequence because 9 has only one middle divisor: 3.
On the other hand, in accordance with the first conjecture, 9 is in the sequence because there are two partitions of 9 into an odd number of consecutive parts: [9], [4, 3, 2], and there is only one partition of 9 into an even number of consecutive parts: [5, 4], therefore the difference of the number of those partitions is 2 - 1 = 1.
On the other hand, in accordance with the second conjecture, 9 is in the sequence because the symmetric representation of sigma(9) = 13 has width 1 on the main diagonal, as shown below in the first quadrant:
.
.     _ _ _ _ _ 5
.    |_ _ _ _ _|
.              |_ _ 3
.              |_  |
.                |_|_ _ 5
.                    | |
.                    | |
.                    | |
.                    | |
.                    |_|
.

Hartmut F. W. Hoft, Proofs of conjectures 2 and 3

Column 1 of A320051.
First differs from A028982 at a(14).
For the definition of middle divisors see A067742.
Cf. A000079, A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320142.
Cf. A016754, A028982, A249223, A259417.

(* computation based on counts of divisors *)
middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#A237048 and A249223 for width at diagonal *)
a249223[n_] := Drop[FoldList[Plus, 0, Map[(-1)^(#+1) a237048[n, #]&, Range[Floor[(Sqrt[8n+1]-1)/2]]]], 1]
a320137W[n_] := Select[Range[n], Last[a249223[#]]==1&]
a320137W[2116]
(* Hartmut F. W. Hoft, May 24 2022 *)

A346868 Sum of divisors of the numbers with no middle divisors.

Original entry on oeis.org

4, 6, 8, 18, 12, 14, 24, 18, 20, 32, 36, 24, 42, 40, 30, 32, 48, 54, 38, 60, 56, 42, 44, 84, 72, 48, 72, 98, 54, 72, 80, 90, 60, 62, 96, 84, 68, 126, 96, 72, 74, 114, 124, 140, 168, 80, 126, 84, 108, 132, 120, 90, 168, 128, 144, 120, 98, 102, 216, 104, 192, 162, 108, 110

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is equal to zero.
So knowing this characteristic shape we can know if a number has middle divisors  (or not) just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with no middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
All terms are even numbers.

			a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
Illustration of initial terms:
m(n) = A071561(n).
.
   n   m(n) a(n)   Diagram
.                      _   _   _     _ _   _ _     _   _   _ _ _     _
                      | | | | | |   | | | | | |   | | | | | | | |   | |
                   _ _|_| | | | |   | | | | | |   | | | | | | | |   | |
   1    3    4    |_ _|  _|_| | |   | | | | | |   | | | | | | | |   | |
                   _ _ _|    _|_|   | | | | | |   | | | | | | | |   | |
   2    5    6    |_ _ _|  _|    _ _| | | | | |   | | | | | | | |   | |
                   _ _ _ _|     |  _ _|_| | | |   | | | | | | | |   | |
   3    7    8    |_ _ _ _|  _ _|_|    _ _|_| |   | | | | | | | |   | |
                            |  _|     |  _ _ _|   | | | | | | | |   | |
                   _ _ _ _ _| |      _|_|    _ _ _|_| | | | | | |   | |
   4   10   18    |_ _ _ _ _ _|  _ _|       |    _ _ _|_| | | | |   | |
   5   11   12    |_ _ _ _ _ _| |  _|      _|   |  _ _ _ _|_| | |   | |
                   _ _ _ _ _ _ _| |      _|  _ _| | |  _ _ _ _|_|   | |
   6   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _ _| |
   7   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|
                                    |  _ _|  _ _|_|       | |
                   _ _ _ _ _ _ _ _ _| |  _ _|  _|        _|_|
   8   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|         |
                   _ _ _ _ _ _ _ _ _ _| |  _ _|        _|
   9   19   20    |_ _ _ _ _ _ _ _ _ _| | |        _ _|
                   _ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
  10   21   32    |_ _ _ _ _ _ _ _ _ _ _| | |  _ _|
  11   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  12   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
                                            | |
                   _ _ _ _ _ _ _ _ _ _ _ _ _| |
  13   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071561(n)).

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 *)

A346867 Sum of divisors of the numbers that have middle divisors.

Original entry on oeis.org

1, 3, 7, 12, 15, 13, 28, 24, 31, 39, 42, 60, 31, 56, 72, 63, 48, 91, 90, 96, 78, 124, 57, 93, 120, 120, 168, 104, 127, 144, 144, 195, 96, 186, 121, 224, 180, 234, 112, 252, 171, 156, 217, 210, 280, 216, 248, 182, 360, 133, 312, 255, 252, 336, 240, 336, 168, 403, 372, 234

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is >= 1.
Also the width on the main diagonal equals the number of middle divisors.
So knowing this characteristic shape we can know if a number has middle divisors (or not) and the number of them just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.

			a(4) = 12 because the sum of divisors of the fourth number that has middle divisors (i.e., 6) is 1 + 2 + 3 + 6 = 12.
On the other hand we can see that in the main diagonal of every diagram the width is >= 1 as shown below.
Illustration of initial terms:
m(n) = A071562(n).
.
   n   m(n) a(n)   Diagram
.                  _ _   _   _   _ _     _     _ _   _   _       _
   1    1    1    |_| | | | | | | | |   | |   | | | | | | |     | |
   2    2    3    |_ _|_| | | | | | |   | |   | | | | | | |     | |
                   _ _|  _|_| | | | |   | |   | | | | | | |     | |
   3    4    7    |_ _ _|    _|_| | |   | |   | | | | | | |     | |
                   _ _ _|  _|  _ _|_|   | |   | | | | | | |     | |
   4    6   12    |_ _ _ _|  _| |  _ _ _| |   | | | | | | |     | |
                   _ _ _ _| |_ _|_|    _ _|   | | | | | | |     | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | |     | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | |     | |
                              |  _ _|    _| |    _ _ _|_| |     | |
                   _ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|     | |
   7   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| |    _ _ _ _ _| |
                                  |  _ _|  _|    _|   |    _ _ _ _|
                   _ _ _ _ _ _ _ _| |     |     |  _ _|   |
   8   15   24    |_ _ _ _ _ _ _ _| |  _ _|  _ _|_|       |
   9   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|
                   _ _ _ _ _ _ _ _ _| | |     |      _|
  10   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|
                   _ _ _ _ _ _ _ _ _ _| | |       |
  11   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
                                          | |
                                          | |
                   _ _ _ _ _ _ _ _ _ _ _ _| |
  12   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _|
.
The n-th diagram has the property that at least it shares a vertex with the (n+1)-st diagram.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A240542, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346868 (of numbers with no middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], Plus @@ d, 0]]; Select[Array[s, 150], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 ; \\ A071562
apply(sigma, select(is, [1..200])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071562(n)).

cp's OEIS Frontend

A299761 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which row n lists the middle divisors of n, or 0 if there are no middle divisors of n.

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A303297 List of middle divisors: for every positive integer that has middle divisors, add its middle divisors to the sequence.

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A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

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A319529 Odd numbers that have middle divisors.

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A319802 Even numbers without middle divisors.

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A319801 Odd numbers without middle divisors.

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A320137 Numbers that have only one middle divisor.

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A346868 Sum of divisors of the numbers with no middle divisors.

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