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A320142 Numbers that have exactly two middle divisors.

%I A320142 #29 Aug 23 2024 20:54:12
%S A320142 6,12,15,20,24,28,30,35,40,42,45,48,54,56,60,63,66,70,77,80,84,88,90,
%T A320142 91,96,99,104,108,110,112,117,126,130,132,135,140,143,150,153,154,156,
%U A320142 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
%N A320142 Numbers that have exactly two middle divisors.
%C A320142 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.
%C A320142 Conjecture 2: numbers k with the property that symmetric representation of sigma(k) has width 2 on the main diagonal.
%C A320142 By the theorem in A067742 conjecture 2 is true. - _Hartmut F. W. Hoft_, Aug 18 2024
%e A320142 15 is in the sequence because 15 has two middle divisors: 3 and 5.
%e A320142 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.
%e A320142 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:
%e A320142 .                                _
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                               | |
%e A320142 .                          _ _ _|_|
%e A320142 .                      _ _| |      8
%e A320142 .                     |    _|
%e A320142 .                    _|  _|
%e A320142 .                   |_ _|  8
%e A320142 .                   |
%e A320142 .    _ _ _ _ _ _ _ _|
%e A320142 .   |_ _ _ _ _ _ _ _|
%e A320142 .                    8
%e A320142 .
%t A320142 a320142Q[k_] := Length[Select[Divisors[k], k/2<=#^2<2k&]]==2
%t A320142 a320142[n_] := Select[Range[n], a320142Q]
%t A320142 a320142[260] (* _Hartmut F. W. Hoft_, Aug 20 2024 *)
%Y A320142 Column 2 of A320051.
%Y A320142 First differs from A001284 at a(19).
%Y A320142 For the definition of middle divisors see A067742.
%Y A320142 Cf. A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320137.
%K A320142 nonn
%O A320142 1,1
%A A320142 _Omar E. Pol_, Oct 06 2018