This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A304231 #21 Nov 04 2022 20:12:22
%S A304231 1,4,6,9,12,15,16,20,24,25,28,30,35,36,40,42,45,48,49,54,56,60,63,64,
%T A304231 66,70,72,77,80,81,84,88,90,91,96,99,100,104,108,110,112,117,120,121,
%U A304231 126,130,132,135,140,143,144,150,153,154,156,160,165,168,169,170
%N A304231 Numbers of the form m*k with m <= k < 2m.
%C A304231 From _Hartmut F. W. Hoft_, Nov 04 2022: (Start)
%C A304231 Three equivalent properties that describe this sequence:
%C A304231 (1) Numbers j satisfying { (m, k) : j = m*k and m <= k < 2*m } != { } -- definition of the sequence.
%C A304231 (2) Numbers j satisfying { d : d | j and sqrt(j/2) < d < sqrt(2*j) } != { } -- stricter than middle divisors.
%C A304231 (3) Numbers j satisfying { d : d | j and d, j/d <= r(j) } != { } -- r(j) = floor((sqrt(8*j+1)-1)/2).
%C A304231 Computations using property (2) are significantly slower than those using properties (1) or (3). (End)
%H A304231 Hartmut F. W. Hoft, <a href="/A304231/a304231.pdf">Proof of the equivalences</a>.
%e A304231 From _Hartmut F. W. Hoft_, Nov 04 2022: (Start)
%e A304231 72 = 2*6^2 is in this sequence since it has divisors 8 and 9 between 6 and 12.
%e A304231 50 = 2*5^2 is not in this sequence since it has no divisors between 5 and 10.
%e A304231 180 = 2^2 * 3^2 * 5 has the 11 divisors 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18 less than or equal to 18 = r(180), but only the 7 divisors 20, 30, 36, 45, 60, 90, 180 greater than 18. Since sqrt(90) < 10 < 12 < 15 < 18 = r(180) < sqrt(360) and 10 < 18 < 20 and 12 < 15 < 24, all three properties stated above are demonstrated. (End)
%t A304231 (* implementation of property (1) *)
%t A304231 a304231[n_] := Module[{list={}, i, j}, For[i=1, i<=Sqrt[n], i++, j=i; While[i j<=n&&j<2i, AppendTo[list, i j]; j++]]; Union[list]]
%t A304231 a304231[170] (* _Hartmut F. W. Hoft_, Nov 04 2022 *)
%o A304231 (Python) sorted(sum([[i*j for j in range(i,2*i)] for i in range(100)], []))
%o A304231 (PARI) isok(n) = fordiv(n, d, if ((d >= n/d) && (d < 2*n/d), return (1))); \\ _Michel Marcus_, May 25 2018
%Y A304231 Slightly more strict than A071562 -- only some terms of the form 2*j^2 are omitted.
%K A304231 nonn,easy
%O A304231 1,2
%A A304231 _Keenan Pepper_, May 08 2018