A066522 Numbers n whose divisors less than or equal to sqrt(n) are consecutive, from 1 up to some number k.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 22, 23, 24, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 60, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157

Offset: 1

Author

Johan G. van der Galien (galien8(AT)zonnet.nl), Jan 05 2002

Keywords

• nonn
• nice
• easy

Comments

The sequence consists of all numbers of the form p or 2p with p prime, along with 1, 8, 12, 18, 24 and 60. Sketch of proof: If k<=2 then n=1 or 8 or p or 2p. If k>2, then one of the numbers k+1, ..., k+4 is == 2 (mod 4); call it m. Then m/2 is an odd number <= k, so m = 2 * (m/2) divides n. Since m is not among 1,2,...,k, it must be greater than sqrt(n), so sqrt(n) < m <= k+4. Also, n is divisible by all positive integers <= k, including k, k-1 and k-2, whose least common multiple is their product divided by 1 or 2. So n >= k(k-1)(k-2)/2. Combining these inequalities implies k<=7 and n<=120.

Changing the definition to use "less than sqrt(n)" doesn't change the sequence. - Stewart Gordon, Sep 27 2011

Examples

			60 = 1*60 = 2*30 = 3*20 = 4*15 = 5*12 = 6*10.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• J. G. van der Galien, The Dawn of Science.

Crossrefs

Cf. A066664 (composite terms); A074964, A000196.

Cf. A161906.

Programs

Haskell

import Data.List (genericLength)
a066522 n = a066522_list !! (n-1)
a066522_list = filter f [1..] where
   f x = genericLength ds == maximum ds where ds = a161906_row x
-- Reinhard Zumkeller, Jun 24 2015, Nov 14 2011

Mathematica

test[n_] := Module[{}, d=Divisors[n]; d=Take[d, Ceiling[Length[d]/2]]; Last[d]==Length[d]]; Select[Range[1, 200], test]
cdQ[n_]:=Module[{d=Union[Differences[Select[Divisors[n],#<=Sqrt[n]&]]]},d=={}||d=={1}]; Select[Range[200],cdQ] (* Harvey P. Dale, Feb 12 2017 *)

PARI

{ n=0; for (m=1, 10^10, d=divisors(m); b=1; for (i=2, ceil(length(d)/2), if (d[i] - d[i-1] > 1, b=0; break)); if (b, write("b066522.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 21 2010

Extensions

Edited by Dean Hickerson, Jan 07 2002