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A304231 Numbers of the form m*k with m <= k < 2m. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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)
LINKS
Hartmut F. W. Hoft, Proof of the equivalences.
EXAMPLE
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)
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 *)
PROG
(Python) sorted(sum([[i*j for j in range(i, 2*i)] for i in range(100)], []))
(PARI) isok(n) = fordiv(n, d, if ((d >= n/d) && (d < 2*n/d), return (1))); \\ Michel Marcus, May 25 2018
CROSSREFS
Slightly more strict than A071562 -- only some terms of the form 2*j^2 are omitted.
Sequence in context: A076083 A094202 A310666 * A007074 A054087 A079255
KEYWORD
nonn,easy
AUTHOR
Keenan Pepper, May 08 2018
STATUS
approved

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Last modified February 29 20:48 EST 2024. Contains 370428 sequences. (Running on oeis4.)