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A320137 Numbers that have only one middle divisor.

%I A320137 #49 Aug 04 2022 16:10:40
%S A320137 1,2,4,8,9,16,18,25,32,36,49,50,64,81,98,100,121,128,162,169,196,200,
%T A320137 225,242,256,289,324,338,361,392,441,484,512,529,578,625,676,722,729,
%U A320137 784,841,882,961,968,1024,1058,1089,1156,1250,1352,1369,1444,1458,1521,1681,1682,1849,1922,1936,2025,2048,2116
%N A320137 Numbers that have only one middle divisor.
%C A320137 Conjecture 1: sequence consists of 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 1.
%C A320137 Conjecture 2: sequence consists of numbers k with the property that the symmetric representation of sigma(k) has width 1 on the main diagonal.
%C A320137 Conjecture 3: all powers of 2 are in the sequence.
%C A320137 From _Hartmut F. W. Hoft_, May 24 2022: (Start)
%C A320137 Every number in this sequence is a square or twice a square, i.e., this sequence is a subsequence of A028982, and conjectures 2 and 3 are true (see the link for proofs). Furthermore, all odd numbers in this sequence are squares and form subsequences of A016754 and of A319529.
%C A320137 Every number k in this sequence has the form k = 2^m * q^2, m >= 0, q >= 1 odd, where for any divisor e of q^2 smaller than the largest divisor of q^2 that is less than or equal to row(q^2) = floor((sqrt(8*q^2 + 1) - 1)/2) the inequalities 2^(m+1) * e < row(n) hold (see the link for a proof).
%C A320137 The smallest odd square not in this sequence is 1225 = 35^2 = (5*7)^2 since it has the 3 middle divisors 25, 35, 49 and the width of the symmetric representation of sigma(1225) at the diagonal equals 3. However, the squares of odd primes in this sequence are a subsequence of A259417.
%C A320137 The smallest even square not in this sequence is 144 = 12^2 = (2*2*3)^2 since it has the 3 middle divisors 9, 12, 16 and the width of the symmetric representation of sigma(144) at the diagonal equals 3.
%C A320137 The smallest twice square not in this sequence is 72 = 2 * (2*3)^2 = 2^3 * 3^2 since it has the 3 middle divisors 6, 8, 9 and the width of the symmetric representation of sigma(72) at the diagonal equals 3.
%C A320137 Apart from the powers of 2 in the infinite first row, the numbers in the sequence can be arranged as an irregular triangle with each row containing the finitely many numbers q^2, 2 * q^2, 4 * q^2, ..., 2^m * q^2 satisfying the condition stated above, as shown below:
%C A320137     1     2     4     8     16     32     64     128     256 ...
%C A320137     9    18    36
%C A320137    25    50   100   200
%C A320137    49    98   196   392    784
%C A320137    81   162   324
%C A320137   121   242   484   968   1936   3872
%C A320137   169   338   676  1352   2704   5408  10816
%C A320137   225
%C A320137   289   578  1156  2312   4624   9248  18496   36992
%C A320137   361   722  1444  2888   5776  11552  23104   46208
%C A320137   441   882
%C A320137   529  1058  2116  4232   8464  16928  33856   67712  135424
%C A320137   625  1250  2500  5000
%C A320137   729  1458  2916
%C A320137   841  1682  3364  6728  13456  26912  53824  107648  215296
%C A320137   ...
%C A320137 (End)
%H A320137 Hartmut F. W. Hoft, <a href="/A320137/a320137.pdf">Proofs of conjectures 2 and 3</a>
%e A320137 9 is in the sequence because 9 has only one middle divisor: 3.
%e A320137 On the other hand, in accordance with the first conjecture, 9 is in the sequence because there are two partitions of 9 into an odd number of consecutive parts: [9], [4, 3, 2], and there is only one partition of 9 into an even number of consecutive parts: [5, 4], therefore the difference of the number of those partitions is 2 - 1 = 1.
%e A320137 On the other hand, in accordance with the second conjecture, 9 is in the sequence because the symmetric representation of sigma(9) = 13 has width 1 on the main diagonal, as shown below in the first quadrant:
%e A320137 .
%e A320137 .     _ _ _ _ _ 5
%e A320137 .    |_ _ _ _ _|
%e A320137 .              |_ _ 3
%e A320137 .              |_  |
%e A320137 .                |_|_ _ 5
%e A320137 .                    | |
%e A320137 .                    | |
%e A320137 .                    | |
%e A320137 .                    | |
%e A320137 .                    |_|
%e A320137 .
%t A320137 (* computation based on counts of divisors *)
%t A320137 middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
%t A320137 a320137D[n_] := Select[Range[n], Length[middleDiv[#]]==1&]
%t A320137 a320137D[2116]
%t A320137 (* computation based on A237048 and A249223 for width at diagonal *)
%t A320137 a249223[n_] := Drop[FoldList[Plus, 0, Map[(-1)^(#+1) a237048[n, #]&, Range[Floor[(Sqrt[8n+1]-1)/2]]]], 1]
%t A320137 a320137W[n_] := Select[Range[n], Last[a249223[#]]==1&]
%t A320137 a320137W[2116]
%t A320137 (* _Hartmut F. W. Hoft_, May 24 2022 *)
%Y A320137 Column 1 of A320051.
%Y A320137 First differs from A028982 at a(14).
%Y A320137 For the definition of middle divisors see A067742.
%Y A320137 Cf. A000079, A071561, A071562, A237048, A237593, A240542, A245092, A249351 (widths), A279286, A279387, A280849, A281007, A299761, A299777, A303297, A319529, A319796, A319801, A319802, A320142.
%Y A320137 Cf. A016754, A028982, A249223, A259417.
%K A320137 nonn
%O A320137 1,2
%A A320137 _Omar E. Pol_, Oct 06 2018