A113966 a(1)=1; for n>1, a(n) is the smallest positive integer not occurring earlier in the sequence such that |a(n)-a(n-1)| does not divide a(n).

1, 3, 5, 2, 6, 10, 4, 7, 9, 11, 8, 13, 15, 17, 12, 19, 14, 18, 22, 16, 21, 23, 20, 26, 29, 24, 31, 25, 27, 32, 35, 33, 28, 34, 30, 37, 39, 41, 36, 43, 38, 42, 46, 40, 47, 44, 49, 45, 51, 53, 48, 55, 52, 57, 50, 54, 58, 61, 56, 59, 62, 65, 63, 67, 60, 68, 71, 64, 69, 73, 66, 70

Offset

1,2

Comments

Sequence is a permutation of the positive integers.

Proof that every number must eventually appear: Suppose not, let k be smallest number that never appears. Then for every a(n-1) > k, we have a(n-1)-k | k, i.e. i(a(n-1)-k) = k for some i with 1 <= i <= k. Therefore a(n-1) <= (k+ik)/k <= k(k+1). So once a(n-1) > k(k+1), we will have a(n) = k, a contradiction. - N. J. A. Sloane, Jan 29 2006

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Example

Among those positive integers not among the first 4 integers of the sequence, a(5) = 6 is the smallest such that |a(5)-a(4)| = |6-2| = 4 does not divide a(5) =6. 4, for example, is not among the first 4 terms of the sequence, but |4-2| = 2 does divide 4. So a(5) is not 4, but is instead 6.

Mathematica

f[l_] := Block[{k=1}, While[MemberQ[l, k] || Mod[k, Abs[k - Last[l]]] == 0, k++ ]; Return[Append[l, k]]; ]; Nest[f, {1}, 100] (* Ray Chandler, Nov 13 2005 *)

Prog

(Haskell)
import Data.List (delete)
a113966 n = a113966_list !! (n-1)
a113966_list = 1 : f [2..] [1] where
   f xs ys'@(y:ys) = y' : f (delete y' xs) (y':ys') where
     y' = head [z | z <- xs, y `mod` abs (z - y) > 0]
-- Reinhard Zumkeller, Feb 26 2013

Crossrefs

Cf. A113963, A113967.

Cf. A222622 (inverse), A222623 (fixed points).

Sequence in context: A333111 A139584 A064790 * A164611 A316086 A227988

Adjacent sequences: A113963 A113964 A113965 * A113967 A113968 A113969

Keyword

nonn,nice

Author

Leroy Quet, Nov 10 2005

Extensions

Extended by Jim Nastos and Ray Chandler, Nov 13 2005

Status

approved