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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.
3
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.
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:
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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:
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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.
Sequence in context: A122053 A124084 A133045 * A341489 A158858 A202356
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Oct 04 2018
STATUS
approved