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%I #44 Apr 09 2023 02:45:30
%S 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 80,84,88,90,96,98,100,104,108,110,112,120,126,128,130,132,140,144,
%U 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 Even numbers that have middle divisors, where "middle divisor" means a divisor in the half-open interval [sqrt(n/2), sqrt(n*2)).
%C Even numbers k such that the symmetric representation of sigma(k) has an odd number of parts.
%C An even number A005843 is in this sequence iff A067742(t) != 0.
%C For the definition of middle divisors, see A067742.
%C For more information about the symmetric representation of sigma(k) see A237593.
%C From _Hartmut F. W. Hoft_, Mar 28 2023: (Start)
%C 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 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 Michael De Vlieger, <a href="/A319796/b319796.txt">Table of n, a(n) for n = 1..10000</a>
%e 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 . _ _ _ _
%e . |_ _ _ |_ 12
%e . | |_
%e . |_ _ |
%e . | |
%e . | |
%e . |_|
%e .
%e 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 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 filter:= n -> ormap(t -> t^2 >= n/2 and t^2 < 2*n, numtheory:-divisors(n)):
%p select(filter, 2*[$1..1000]); # _Robert Israel_, Mar 29 2023
%t middleDiv[n_] := Select[Divisors[n], Sqrt[n/2]<=#<Sqrt[2n]&]
%t a319796[n_] := Select[Range[2, n, 2], middleDiv[#]!={}&]
%t a319796[256] (* _Hartmut F. W. Hoft_, Mar 28 2023 *)
%Y Intersection of A005843 and A071562.
%Y Cf. A000203, A067742, A071090, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319801, A319802.
%K nonn
%O 1,1
%A _Omar E. Pol_, Sep 28 2018
%E Name clarified by _Omar E. Pol_, Mar 28 2023
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