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A084937 Smallest number which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2. 29
1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 15, 29, 14, 25, 27, 22, 31, 35, 12, 37, 41, 18, 43, 47, 20, 33, 49, 26, 45, 53, 28, 39, 55, 32, 51, 59, 38, 61, 63, 34, 65, 57, 44, 67, 69, 40, 71, 73, 24, 77, 79, 30, 83, 89, 36, 85, 91, 46, 75, 97, 52, 81 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Equivalently, this is the lexicographically earliest sequence of positive numbers satisfying the condition that each term is relatively prime to the next two terms. - N. J. A. Sloane, Nov 03 2014

All primes and prime powers occur, and the primes occur in their natural order. For any prime p, p occurs before p^2 before p^3, ...

Empirically, this is a permutation of natural numbers, with inverse A084933: a(A084933(n))=A084933(a(n))=n. It seems that there are no further fixed points after {1,2,3,8,33,39}. Empirically, a(n) mod 2 = A011655(n+1); ABS(a(n)-n) < n; a(3*n+1)>n; a(3*n+2)<n. - Reinhard Zumkeller, Dec 16 2007

For a(n) mod 3 see A249603. - N. J. A. Sloane, Nov 03 2014

A249694(n) = GCD(a(n),a(n+3)). - Reinhard Zumkeller, Nov 04 2014

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..100000

John P. Linderman, Table of n, a(n) for n = 1..1000000 (about 14MB)

Rémy Sigrist, Scatterplot of the first 2500 terms

Index entries for sequences that are permutations of the natural numbers

FORMULA

Empirically, the points lie roughly on two lines: if n == 2 mod 3 then a(n) ~= 2n/3, otherwise a(n) ~= 4n/3. See A249680-A249683 for the three trisections. - N. J. A. Sloane, Nov 03 2014, Nov 04 2014

MAPLE

N:= 1000: # to get a(n) until the first entry > N

a[1]:= 1: a[2]:= 2:

R:= {$3..N}:

for n from 3 while R <> {} do

  success:= false;

  for r in R do

    if igcd(r, a[n-1]) = 1 and igcd(r, a[n-2])=1 then

       a[n]:= r;

       R:= R minus {r};

       success:= true;

       break

    fi

  od:

  if not success then break fi;

od:

seq(a[i], i = 1 .. n-1); # Robert Israel, Dec 12 2014

MATHEMATICA

lst={1, 2, 3}; unused=Range[4, 100]; While[n=Select[unused, CoprimeQ[#, lst[[-1]]] && CoprimeQ[#, lst[[-2]]] &, 1]; n != {}, AppendTo[lst, n[[1]]]; unused=DeleteCases[unused, n[[1]]]]; lst

f[s_] := Block[{k = 1, l = Take[s, -2]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Nest[f, {1, 2}, 67] (* Robert G. Wilson v, Jun 26 2011 *)

PROG

(Haskell)

import Data.List (delete)

a084937 n = a084937_list !! (n-1)

a084937_list = 1 : 2 : f 2 1 [3..] where

   f x y zs = g zs where

      g (u:us) | gcd y u > 1 || gcd x u > 1 = g us

               | otherwise = u : f u x (delete u zs)

-- Reinhard Zumkeller, Jan 28 2012

(Python)

from fractions import gcd

A084937_list, l1, l2, s, b = [1, 2], 2, 1, 3, set()

for _ in range(10**3):

....i = s

....while True:

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

............A084937_list.append(i)

............l2, l1 = l1, i

............b.add(i)

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

................b.remove(s)

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

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

........i += 1 # Chai Wah Wu, Dec 09 2014

(PARI) taken(k, t=v[k])=for(i=3, k-1, if(v[i]==t, return(1))); 0

step(k, g)=while(gcd(k, g)>1, k++); k

first(n)=local(v=vector(n, i, i)); my(nxt=3, t); for(k=3, n, v[k]=step(nxt, t=v[k-1]*v[k-2]); while(taken(k), v[k]=step(v[k]+1, t)); if(v[k]==t, while(taken(k+1, t++), ))); v \\ Charles R Greathouse IV, Aug 26 2016

CROSSREFS

Cf. A084933 (inverse), A103683, A121216, A247665, A090252, A249603 (read mod 3), A249680, A249681, A249682, A249683 (trisections), A249694, A011655, A249684 (numbers that take a record number of steps to appear), A249685.

Indices of primes: A249602, and of prime powers: A249575.

Running counts of missing numbers: A249686, A250099, A250100; A249777, A249856, A249857.

Where a(3n)>a(3n+1): A249689.

Sequence in context: A118318 A245707 A271861 * A269367 A081994 A249064

Adjacent sequences:  A084934 A084935 A084936 * A084938 A084939 A084940

KEYWORD

nonn,look

AUTHOR

Reinhard Zumkeller, Jun 13 2003

EXTENSIONS

Entry revised by N. J. A. Sloane, Nov 04 2014

STATUS

approved

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Last modified November 21 22:32 EST 2017. Contains 295054 sequences.