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A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).

%I A319796 #44 Apr 09 2023 02:45:30
%S A319796 2,4,6,8,12,16,18,20,24,28,30,32,36,40,42,48,50,54,56,60,64,66,70,72,
%T A319796 80,84,88,90,96,98,100,104,108,110,112,120,126,128,130,132,140,144,
%U A319796 150,154,156,160,162,168,170,176,180,182,190,192,196,198,200,204,208,210,216,220,224,228,234,238,240,242,252,256
%N A319796 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).
%C A319796 Even numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
%C A319796 An even number A005843 is in this sequence iff A067742(t) != 0.
%C A319796 For the definition of middle divisors, see A067742.
%C A319796 For more information about the symmetric representation of sigma(k) see A237593.
%C A319796 From _Hartmut F. W. Hoft_, Mar 28 2023: (Start)
%C A319796 By Theorem 1 (iii) in A067742, the number of middle divisors of a(n) equals the width of the symmetric representation of sigma(a(n)), SRS(a(n)), on the diagonal which equals the triangle entry A249223(n, A003056(n)). The maximum widths of the center part of SRS(a(n)) need not occur at the diagonal.
%C A319796 For example, a(7) = 2 * 3^2 = 18, SRS(18) has a single part with maximum width 2 while its width at the diagonal equals 1 = A067742(18), and divisor 3 is the only middle divisor of a(7). (End)
%H A319796 Michael De Vlieger, <a href="/A319796/b319796.txt">Table of n, a(n) for n = 1..10000</a>
%e A319796 6 is in the sequence because it's an even number and the symmetric representation of sigma(6) = 12 has an odd number of parts (more exactly only one part), as shown below:
%e A319796 .    _ _ _ _
%e A319796 .   |_ _ _  |_ 12
%e A319796 .         |   |_
%e A319796 .         |_ _  |
%e A319796 .             | |
%e A319796 .             | |
%e A319796 .             |_|
%e A319796 .
%e A319796 Also 50 is in the sequence because it's an even number and the symmetric representation of sigma(50) = 93 has an odd number of parts (more exactly three parts), they are [39, 15, 39].
%e A319796 a(34) = 110 = 2 * 5 * 11 has 10 and 11 as its middle divisors, and SRS(a(34)) has 3 parts and width 2 at the diagonal. -  _Hartmut F. W. Hoft_, Mar 28 2023
%p A319796 filter:= n -> ormap(t -> t^2 >= n/2 and t^2 < 2*n, numtheory:-divisors(n)):
%p A319796 select(filter, 2*[$1..1000]); # _Robert Israel_, Mar 29 2023
%t A319796 middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
%t A319796 a319796[n_] := Select[Range[2, n, 2], middleDiv[#]!={}&]
%t A319796 a319796[256] (* _Hartmut F. W. Hoft_, Mar 28 2023 *)
%Y A319796 Intersection of A005843 and A071562.
%Y A319796 Cf. A000203, A067742, A071090, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319801, A319802.
%K A319796 nonn
%O A319796 1,1
%A A319796 _Omar E. Pol_, Sep 28 2018
%E A319796 Name clarified by _Omar E. Pol_, Mar 28 2023