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A098550 The Yellowstone permutation: a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1). 121
1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17, 18, 85, 24, 55, 34, 65, 36, 91, 30, 49, 38, 63, 19, 42, 95, 44, 57, 40, 69, 50, 23, 48, 115, 52, 75, 46, 81, 56, 87, 62, 29, 31, 58, 93, 64, 99, 68, 77, 54, 119, 60 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For n>3, GCD(a(n),a(n-1))=1 and GCD(a(n),a(n-2))>1. (This is just a restatement of the definition.)

This is now known to be a permutation of the natural numbers: see the 2015 article by Applegate, Havermann, Selcoe, Shevelev, Sloane, and Zumkeller,

From N. J. A. Sloane, Nov 28 2014: (Start)

Some of the known properties (but see the above-mentioned article for a fuller treatment):

1. The sequence is infinite. Proof: We can always take a(n) = a(n-2)*p, where p is a prime that is larger than any prime dividing a(1), ..., a(n-1). QED

2. At least one-third of the terms are composite. Proof: The sequence cannot contain three consecutive primes. So at least one term in three is composite. QED

3. For any prime p, there is a term that is  divisible by p. Proof: Suppose not. (i) No prime q>p can divide any term. For if a(n)=kq is the first multiple of q to appear, then we could have used kp<kq instead, a contradiction. So every term a(n) is a product of primes < p. (ii) Choose N such that a(n) > p^2 for all n > N. For n > N, let a(n)=bg, a(n+1)=c, a(n+2)=dg, where g=gcd(a(n),a(n+2)). Let q be the largest prime factor of g. We know q<p, so qp < p^2 < dg, so we could have used qp instead of dg, a contradiction. QED

3a. Let a(n_p) be the first term that is divisible by p (this is A251541). Then a(n_p) = q*p where q is a prime less than p. If p < r are primes then n_p < n_r. Proof: Immediate consequences of the definition.

4. (From David Applegate, Nov 27 2014) There are infinitely many even terms. Proof:

Suppose not.  Then let 2x be the maximum even entry.  Because the sequence is infinite, there exists an N such that for any n > N, a(n) is odd, and a(n) > x^2.

In addition, there must be some n > N such that a(n) < a(n+2).  For that n, let g = gcd(a(n),a(n+2)), a(n) = bg, a(n+1)=c, a(n+2)=dg, with all of b,c,d,g relatively prime, and odd.

Since dg > bg, d > b >= 1, so d >= 3.  Also, g >= 3.

Since a(n) = bg > x^2, one of b or g is > x.

Case 1: b > x.  Then 2b > 2x, so 2b has not yet occurred in the sequence.  And gcd(bg,2b)=b > x > 1, gcd(2b,c)=1, and since g >= 3, 2b < bg < dg.  So a(n+2) should have been 2b instead of dg.

Case 2: g > x.  Then 2g > 2x, so 2g has not yet occurred in the sequence.  And gcd(bg,2g)=g > 1, gcd(2g,c)=1, and since d >= 3, 2g < dg.  So a(n+2) should have been 2g instead of dg.

In either case, we derive a contradiction. QED

Conjectures:

5. For any prime p > 97, the first time we see p, it is in the subsequence a(n) = 2b, a(n+2) = 2p, a(n+4) = p for some n, b, where n is about 2.14*p and gcd(b,p)=1.

6. The value of |{k=1,..,n: a(k)<=k}|/n tends to 1/2. - Jon Perry, Nov 22 2014 [Comment edited by N. J. A. Sloane, Nov 23 2014 and Dec 26 2014]

7. Based on the first 250000 terms, I conjectured on Nov 30 2014 that a(n)/n <= (Pi/2)*log n.

8. The primes in the sequence appear in their natural order. This conjecture is very plausible but as yet there is no proof. - N. J. A. Sloane, Jan 29 2015

(End)

The only fixed points seem to be {1, 2, 3, 4, 12, 50, 86} - see A251411. Checked up to n=10^4. - L. Edson Jeffery, Nov 30 2014. No further terms up to 10^5 - M. F. Hasler, Dec 01 2014; up to 250000 - Reinhard Zumkeller; up to 300000 (see graph) - Hans Havermann, Dec 01 2014; up to 10^6 - Chai Wah Wu, Dec 06 2014; up to 10^8 - David Applegate, Dec 08 2014.

From N. J. A. Sloane, Dec 04 2014: (Start)

The first 250000 points lie on about 8 roughly straight lines, whose slopes are approximately 0.467, 0.957, 1.15, 1.43, 2.40, 3.38, 5.25 and 6.20.

The first six lines seem well-established, but the two lines with highest slope at present are rather sparse. Presumably as the number of points increases, there will be more and more lines of ever-increasing slopes.

These lines can be seen in the Havermann link. See the "slopes" link for a list of the first 250000 terms sorted according to slope (the four columns in the table give n, a(n), the slope a(n)/n, and the number of divisors of a(n), respectively).

The primes (with two divisors) all lie on the lowest line, and the lines of slopes 1.43 and higher essentially consist of the products of two primes (with four divisors).

(End)

The eight roughly straight lines mentioned above are actually curves.  A good fit for the "line" with slope ~= 1.15 is a(n)~=n(1+1.0/log(n/24.2)), and a good fit for the other "lines" is a(n)~= (c/2)*n(1-0.5/log(n/3.67)), for c = 1,2,3,5,7,11,13.  The first of these curves consists of most of the odd terms in the sequence.  The second family consists of the primes (c=1), even terms (c=2), and c*prime (c=3,5,7,11,13,...).  This functional form for the fit is motivated by the observed pattern (after the first 204 terms) of alternating even and odd terms, except for the sequence pattern 2*p, odd, p, even, q*p when reaching a prime (with q a prime < p). - Jon E. Schoenfield and David Applegate, Dec 15 2014

For a generalization, see the sequence of monomials of primes in the comment in A247225. - Vladimir Shevelev, Jan 19 2015

From Vladimir Shevelev, Feb 24 2015: (Start)

Let P be prime. Denote by S_P*P the first multiple of P appearing in the sequence. Then

1) For P>=5, S_P is prime.

Indeed, let

a(n-2)=v, a(n-1)=w, a(n)=S_P*P.     (*)

Note that GCD(v,P)=1. Therefore, by the definition of the sequence, S_P*P should be the smallest number such that GCD(v,S_P)>1.

So S_P is the smallest prime factor of v.

2) The first multiples of all primes appear in the natural order.

Suppose not. Then there is a pair of primes P<Q such that S_Q*Q appears earlier than S_P*P. Let

a(m-2)=v_1, a(m-1)=w_1, a(m)=S_Q*Q.    (**)

Then, as in (*), S_Q is the smallest prime factor of v_1. But this does not depend on Q. So S_Q*P is a smaller candidate in (**), a contradiction.

3) S_P < P.

Indeed, from (*) it follows that the first multiple of S_P appears earlier than the first multiple of P. So, by 2), S_P < P.

(End)

For any given set S of primes, the subsequence consisting of numbers whose prime factors are exactly the primes in S appears in increasing order. For example, if S = {2,3}, 6 appears first, in due course followed by 12, 18, 24, 36, 48, 54, 72, etc. The smallest numbers in each subsequence (i.e., those that appear first) are the squarefree numbers A005117(n), n>1. - Bob Selcoe, Mar 06 2015

LINKS

Paul Tek, Table of n, a(n) for n = 1..10000

David Applegate, C++ program for efficiently computing terms

David Applegate, Revised C++ program with option to print more variables

David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015. Also Journal of Integer Sequences, Vol. 18 (2015), Article 15.6.7

Hans Havermann, Graph of first 300000 terms (the red line is y=x)

Hans Havermann, Loops and unresolved chains for map n -> A098550(n) trajectories

L. Edson Jeffery, Log plot of first 2484 terms.

N. J. A. Sloane, First 250000 terms sorted according to slope a(n)/n

Reinhard Zumkeller, Table of n, a(n) for n = 1..250000

Index entries for sequences that are permutations of the natural numbers

MAPLE

N:= 10^4: # to get a(1) to a(n) where a(n+1) is the first term > N

B:= Vector(N, datatype=integer[4]):

for n from 1 to 3 do A[n]:= n: od:

for n from 4 do

  for k from 4 to N do

    if B[k] = 0 and igcd(k, A[n-1]) = 1 and igcd(k, A[n-2]) > 1 then

       A[n]:= k;

       B[k]:= 1;

       break

    fi

  od:

  if k > N then break fi

od:

seq(A[i], i=1..n-1); # Robert Israel, Nov 21 2014

MATHEMATICA

f[lst_List] := Block[{k = 4}, While[ GCD[ lst[[-2]], k] == 1 || GCD[ lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]]; Nest[f, {1, 2, 3}, 68] (* Robert G. Wilson v, Nov 21 2014 *)

NN = Range[4, 1000]; a098550 = {1, 2, 3}; g = {-1}; While[g[[1]] != 0, g = Flatten[{FirstPosition[NN, v_ /; GCD[a098550[[-1]], v] == 1 && GCD[a098550[[-2]], v] > 1, 0]}]; If[g[[1]] != 0, d = NN[[g]]; a098550 = Flatten[Append[a098550, d[[1]]]]; NN = Delete[NN, g[[1]]]]]; Table[a098550[[n]], {n, 71}] (* L. Edson Jeffery, Jan 01 2015 *)

PROG

(Haskell)

import Data.List (delete)

a098550 n = a098550_list !! (n-1)

a098550_list = 1 : 2 : 3 : f 2 3 [4..] where

   f u v ws = g ws where

     g (x:xs) = if gcd x u > 1 && gcd x v == 1

                   then x : f v x (delete x ws) else g xs

-- Reinhard Zumkeller, Nov 21 2014

(PARI) a(n, show=1, a=3, o=2, u=[])={n<3&&return(n); show&&print1("1, 2"); for(i=4, n, show&&print1(", "a); u=setunion(u, Set(a)); while(#u>1 && u[2]==u[1]+1, u=vecextract(u, "^1")); for(k=u[1]+1, 9e9, gcd(k, o)>1||next; setsearch(u, k)&&next; gcd(k, a)==1||next; o=a; a=k; break)); a} \\ Replace "show" by "a+1==i" in the main loop to print only fixed points. - M. F. Hasler, Dec 01 2014

(Python)

from fractions import gcd

A098550_list, l1, l2, s, b = [1, 2, 3], 3, 2, 4, {}

for _ in range(1, 10**6):

....i = s

....while True:

........if not i in b and gcd(i, l1) == 1 and gcd(i, l2) > 1:

............A098550_list.append(i)

............l2, l1, b[i] = l1, i, 1

............while s in b:

................b.pop(s)

................s += 1

............break

........i += 1 # Chai Wah Wu, Dec 04 2014

CROSSREFS

Cf. A098548, A098551, A249943 (first time all 1..n appear), A251553.

The inverse permutation is in A098551.

A098552(n) = a(a(n)).

A251102(n) = GCD(a(n+2),a(n)).

Cf. A251101 (smallest prime factor), A251103 (largest prime factor), A251138 (number of distinct prime factors), A251140 (total number of prime factors), A251045 (squarefree part), A251089 (squarefree kernel), A250127 and A251415 (records for a(n)/n), A251411 (fixed points), A248647 (records).

Cf. also A251412 (trajectory of 11), A251556 (finite cycles), A251413 and A251414 (variant involving odd numbers), A249357 ("Fibonacci" variant).

Smallest missing numbers: A251416, A251417, A251546-A251552, A247253. See also A251557, A241558, A251559.

Indices of some pertinent subsequences: A251237 (even numbers), A251238 (odd numbers), A251391 (squarefree), A251541 and A251239 (primes), A251240 (prime squares), A251241 (prime powers), A251393 (powers of 2), A251392 (nonprimes), A253297 (primes p for which some multiple k*p > 2*p precedes p).

Three arrays concerning the occurrences of multiples of primes: A251637, A251715, A251716.

Two sequences related to the numbers which immediately follow a prime: A253048, A253049. Seven sequences related to the numbers that appear two steps after primes: A251542, A251543, A251544, A251545, A253052, A253053, A253054. See also A253055 and A253056.

Other starting values: A251554, A251555.

See also A251756, A253297, A251662, A253572, A253573, A253591, A253593, A253588, A253590, A253609, A252865, A252867, A252868, A247225, A247942, A254003, A254077, A254669, A254670, A255509 (version with a priority for appearance of the primes), A255615, A255617, A256189, A256224, A256368, A256461.

See also A064413 (EKG sequence), A255582, A121216 (similar sequences), A257112 (two-dimensional analog).

See also A253765 and A253766 (bisections), A250299 (parity), A253768 (partial sums).

Sequence in context: A115305 A210747 A247942 * A256224 A255509 A257862

Adjacent sequences:  A098547 A098548 A098549 * A098551 A098552 A098553

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Sep 14 2004

STATUS

approved

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Last modified May 28 10:34 EDT 2017. Contains 287240 sequences.