A166133 After initial 1,2,4, a(n+1) is the smallest divisor of a(n)^2-1 that has not yet appeared in the sequence.

1, 2, 4, 3, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 21, 20, 19, 18, 17, 24, 23, 22, 69, 28, 27, 26, 25, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 201, 80, 79

Offset: 1

Author

Franklin T. Adams-Watters, Oct 07 2009

Keywords

• nonn
• nice
• hear

Comments

The initial 1,2,4 provides the smallest example with this rule that is not simply the integers in order, nor (apparently) ends with all divisors of a(n)^2-1 already present.

Apparently the sequence is infinite and includes every positive integer.

Apr 05 2015: John Mason has computed the first ten million terms. See link to zipped file. - N. J. A. Sloane, Apr 06 2015

The sequence contains many runs of incrementing and decrementing values. In the 1200 steps following the 4, there are 136 increments, 706 decrements, and 358 larger steps. What is the limiting distribution for these steps? [Click the "listen" button to appreciate these runs. - N. J. A. Sloane, Apr 03 2015]

After 3, 198, 270, 570, 522, 600, 822, and 882, we have a(n+1) = a(n)^2-1. Does this happen infinitely often? Cf. A256406, A256407.

A256543 gives numbers m such that a(m+1) = a(m)-1 or a(m+1) = a(m)+1. - Reinhard Zumkeller, Apr 01 2015

If this is a permutation, then A255833 is the inverse permutation. - M. F. Hasler, Apr 01 2015

a(A256703(n)+1) = a(A256703(n))^2 - 1. - Reinhard Zumkeller, Apr 08 2015

For n > 3: a(n) = A027750(a(n-1)^2-1, A256751(n)). - Reinhard Zumkeller, Apr 09 2015

Examples

			After a(24) = 22, the divisors of 22^2-1 = 483 are 1, 3, 7, 21, 23, 69, 161, and 483; 1, 3, 7, 21, and 23 have already occurred, so a(25) = 69.
		

Links

• Franklin T. Adams-Watters and N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1203 terms from Franklin T. Adams-Watters)
• Hans Havermann, Log plot of 450000+ terms [Produced by Mathematica's ListLogPlot command]
• Hans Havermann, Over-sized point-joined (over 250000 terms) graph of the sequence [Heavily clipped, which explains the strange appearance. - _N. J. A. Sloane_, Apr 01 2015]
• John Mason, Table of n, a(n) for n = 1..711888 [10 megabytes]
• John Mason, Table of n, a(n) for n = 1..2000000 [32 megabytes]
• John Mason, Ten million terms (zipped file)
• John Mason, Java program to generate this sequence, used to generate 10M terms, and some other associated sequences; it requires splitting into single classes for use.
• N. J. A. Sloane and others, "Blog" about A166133

Crossrefs

Cf. A166134, A000027, A122280, A005563, A256406, A256407, A027750, A005563, A256557, A256559.

For records see A256403, A256404.

Smallest missing numbers: A256405, A256408, A256409.

Cf. A256541 (first differences), A256543.

Inverse (conjectured): A255833.

Cf. A256564 (smallest prime factors), A244080 (largest prime factors), A256578 (largest proper divisors), A256542 (number of divisors).

Upper envelope: the sequence of pairs (A256422(n),A256423(n)).

Cf. A256703.

Cf. A256751.

Programs

Haskell

import Data.List (delete); import Data.List.Ordered (isect)
a166133 n = a166133_list !! (n-1)
a166133_list = 1 : 2 : 4 : f (3:[5..]) 4 where
   f zs x = y : f (delete y zs) y where
            y = head $ isect (a027750_row' (x ^ 2 - 1)) zs
-- Reinhard Zumkeller, Apr 01 2015

Mathematica

s = {1, 2, 4}; e = 4; Do[d = Divisors[e^2 - 1]; i = 1;
While[MemberQ[s, d[[i]]], i++]; e = d[[i]]; AppendTo[s, e], {19997}]; s (* Hans Havermann, Apr 03 2015 *)

PARI

al(n,m=4,u=6)={local(ds,db);
u=bitor(u,1<