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
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:
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. _ _| | 8
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. |_ _| 8
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. _ _ _ _ _ _ _ _|
. |_ _ _ _ _ _ _ _|
. 8
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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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Oct 06 2018
STATUS
approved