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A320142
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Numbers that have exactly two middle divisors.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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:
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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