OFFSET
1,2
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
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..1000
J. G. van der Galien, The Dawn of Science.
EXAMPLE
60 = 1*60 = 2*30 = 3*20 = 4*15 = 5*12 = 6*10.
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Johan G. van der Galien (galien8(AT)zonnet.nl), Jan 05 2002
EXTENSIONS
Edited by Dean Hickerson, Jan 07 2002
STATUS
approved