A304231 Numbers of the form m*k with m <= k < 2m.

1, 4, 6, 9, 12, 15, 16, 20, 24, 25, 28, 30, 35, 36, 40, 42, 45, 48, 49, 54, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160, 165, 168, 169, 170

Offset: 1

Author

Keenan Pepper, May 08 2018

Keywords

• nonn
• easy

Comments

From Hartmut F. W. Hoft, Nov 04 2022: (Start)

Three equivalent properties that describe this sequence:

(1) Numbers j satisfying { (m, k) : j = m*k and m <= k < 2*m } != { } -- definition of the sequence.

(2) Numbers j satisfying { d : d | j and sqrt(j/2) < d < sqrt(2*j) } != { } -- stricter than middle divisors.

(3) Numbers j satisfying { d : d | j and d, j/d <= r(j) } != { } -- r(j) = floor((sqrt(8*j+1)-1)/2).

Computations using property (2) are significantly slower than those using properties (1) or (3). (End)

Examples

			From _Hartmut F. W. Hoft_, Nov 04 2022: (Start)
72 = 2*6^2 is in this sequence since it has divisors 8 and 9 between 6 and 12.
50 = 2*5^2 is not in this sequence since it has no divisors between 5 and 10.
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)
		

Links

• Hartmut F. W. Hoft, Proof of the equivalences.

Crossrefs

Slightly more strict than A071562 -- only some terms of the form 2*j^2 are omitted.

Programs

Mathematica

(* implementation of property (1) *)
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]]
a304231[170] (* Hartmut F. W. Hoft, Nov 04 2022 *)

PARI

isok(n) = fordiv(n, d, if ((d >= n/d) && (d < 2*n/d), return (1))); \\ Michel Marcus, May 25 2018

Python

sorted(sum([[i*j for j in range(i,2*i)] for i in range(100)], []))