A084937 - OEIS

A084937

Smallest number which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2.

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

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, and see also the Sigrist scatterplot. - N. J. A. Sloane, Nov 03 2014, Nov 04 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 the 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..764179 (about 10MB)

Rémy Sigrist, Scatterplot of the first 2500 terms

Index entries for sequences that are permutations of the natural numbers

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

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.