A166133 - OEIS

%I #104 May 30 2023 12:00:26

%S 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,

%T 27,26,25,39,38,37,36,35,34,33,32,31,30,29,40,41,42,43,44,45,46,47,48,

%U 49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,201,80,79

%N 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.

%C 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.

%C Apparently the sequence is infinite and includes every positive integer.

%C Apr 05 2015: _John Mason_ has computed the first ten million terms. See link to zipped file. - _N. J. A. Sloane_, Apr 06 2015

%C 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]

%C 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.

%C A256543 gives numbers m such that a(m+1) = a(m)-1 or a(m+1) = a(m)+1. - _Reinhard Zumkeller_, Apr 01 2015

%C If this is a permutation, then A255833 is the inverse permutation. - _M. F. Hasler_, Apr 01 2015

%C a(A256703(n)+1) = a(A256703(n))^2 - 1. - _Reinhard Zumkeller_, Apr 08 2015

%C For n > 3: a(n) = A027750(a(n-1)^2-1, A256751(n)). - _Reinhard Zumkeller_, Apr 09 2015

%H Franklin T. Adams-Watters and N. J. A. Sloane, <a href="/A166133/b166133.txt">Table of n, a(n) for n = 1..20000</a> (first 1203 terms from Franklin T. Adams-Watters)

%H Hans Havermann, <a href="/A166133/a166133.png">Log plot of 450000+ terms</a> [Produced by Mathematica's ListLogPlot command]

%H Hans Havermann, <a href="http://chesswanks.com/num/oddlybehaved.png">Over-sized point-joined (over 250000 terms) graph of the sequence</a> [Heavily clipped, which explains the strange appearance. - _N. J. A. Sloane_, Apr 01 2015]

%H John Mason, <a href="/A166133/a166133_1.txt">Table of n, a(n) for n = 1..711888</a> [10 megabytes]

%H John Mason, <a href="/A166133/a166133_2.txt">Table of n, a(n) for n = 1..2000000</a> [32 megabytes]

%H John Mason, <a href="https://oeis.org/A166133/a166133-10M.zip">Ten million terms (zipped file)</a>

%H John Mason, <a href="/A166133/a166133.java.txt">Java program to generate this sequence, used to generate 10M terms, and some other associated sequences</a>; it requires splitting into single classes for use.

%H N. J. A. Sloane and others, <a href="/A166133/a166133.txt">"Blog" about A166133</a>

%e 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.

%t s = {1, 2, 4}; e = 4; Do[d = Divisors[e^2 - 1]; i = 1;

%t While[MemberQ[s, d[[i]]], i++]; e = d[[i]]; AppendTo[s, e], {19997}]; s (* _Hans Havermann_, Apr 03 2015 *)

%o (PARI) al(n,m=4,u=6)={local(ds,db);

%o u=bitor(u,1<<m);print1(m);

%o for(i=1,n,

%o ds=divisors(m^2-1);

%o for(k=2,#ds,m=ds[k];db=1<<m;if(!bitand(u,db),break));

%o u=bitor(u,db);print1(","m))}

%o /* This prints the sequence without the initial 1,2. */

%o (Haskell)

%o import Data.List (delete); import Data.List.Ordered (isect)

%o a166133 n = a166133_list !! (n-1)

%o a166133_list = 1 : 2 : 4 : f (3:[5..]) 4 where

%o    f zs x = y : f (delete y zs) y where

%o             y = head $ isect (a027750_row' (x ^ 2 - 1)) zs

%o -- _Reinhard Zumkeller_, Apr 01 2015

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

%Y For records see A256403, A256404.

%Y Smallest missing numbers: A256405, A256408, A256409.

%Y Cf. A256541 (first differences), A256543.

%Y Inverse (conjectured): A255833.

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

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

%Y Cf. A256703.

%Y Cf. A256751.

%K nonn,nice,hear

%O 1,2

%A _Franklin T. Adams-Watters_, Oct 07 2009