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.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..64.

Index entries for sequences that are permutations of the natural numbers

Example

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:
.    _ _ _ _
.   |_ _ _  |_
.         |   |_
.         |_ _  |
.             | |
.             | |
.             |_|
.

Crossrefs

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.

Sequence in context: A122053 A124084 A133045 * A341489 A158858 A202356

Adjacent sequences: A320048 A320049 A320050 * A320052 A320053 A320054

Keyword

nonn,tabl

Author

Omar E. Pol, Oct 04 2018

Status

approved