A320142 Numbers that have exactly two middle divisors.

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..67.

Example

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
.

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A315618 A129494 A001284 * A063931 A001283 A106430

Adjacent sequences: A320139 A320140 A320141 * A320143 A320144 A320145

Keyword

nonn

Author

Omar E. Pol, Oct 06 2018

Status

approved