A123882 a(1)=1, a(2)=2. a(n) = the smallest positive integer that does not occur earlier in the sequence and is coprime to the second-largest term occurring earlier in the sequence.

1, 2, 3, 5, 4, 7, 6, 11, 8, 9, 10, 13, 12, 17, 14, 15, 16, 19, 18, 23, 20, 21, 22, 25, 24, 29, 26, 27, 28, 31, 30, 37, 32, 33, 34, 35, 36, 41, 38, 39, 40, 43, 42, 47, 44, 45, 46, 49, 48, 53, 50, 51, 52, 55, 54, 59, 56, 57, 58, 61, 60, 67, 62, 63, 64, 65, 66, 71, 68, 69, 70, 73

Offset

1,2

Comments

Sequence is a permutation of the positive integers.

From Bob Selcoe, Aug 12 2015: (Start)

Let z be the largest term already occurring in the sequence. Except for a(2), z is odd.

When n=z, the sequence is a permutation of the positive integers up to and including z.

Let p be the smallest prime number that is not a factor of z-1. When n>=3, z+1 is coprime to both z and z-1+p.

When n>=5, a(n) is the smallest positive integer not yet having occurred in the sequence that is coprime to a(n-1). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Bob Selcoe, Aug 12 2015: (Start)

For n>=5, where z and p are defined in "Comments":

a(n) = n-1, except for a(z+1) = z-1+p.

a(n) = a(n-1)+1, except for a(z+1) = a(z)+p and a(z+2) = a(z)+2 (here, p can also be defined as the smallest prime that is not a factor of a(z)). (End)

Example

The second-largest integer among the first 8 terms of the sequence is 7. Those positive terms not occurring among the first 8 terms form the sequence 8,9,10,12,13,...; of these, 8 is the smallest that is coprime to 7. So a(9)=8.
z=43: a(44)=47 because the smallest prime not a factor of z-1 = 42 is 5, and 42+5 = 47. - Bob Selcoe, Aug 12 2015

Maple

N:= 1000: # to get all terms before the first term > N
Next:= Vector(N, i->i+1): Next[N]:= 0:
Prev:= Vector(N, i->i-1):
First:= 3: Prev[3]:= 0:
A[1]:= 1: A[2]:= 2:
Largest:= 2: Second:= 1:
for n from 3 do
   p:= First;
   while igcd(p, Second) > 1 and p <> 0 do
     p:= Next[p];
   od:
   if p = 0 then break fi;
   A[n]:= p;
   if Next[p] <> 0 then Prev[Next[p]]:= Prev[p] fi;
   if p = First then First:= Next[p] else Next[Prev[p]]:= Next[p] fi;
   if p > Largest then
     Second:= Largest; Largest:= p
   elif p > Second then
     Second:= p
   fi
od:
seq(A[k], k=1..n-1); # Robert Israel, Aug 21 2015

Mathematica

f[l_List] := Block[{s = Sort[l][[ -2]], k = 1}, While[GCD[k, s] > 1 || MemberQ[l, k], k++ ]; Append[l, k]]; Nest[f, {1, 2}, 75] (* Ray Chandler, Oct 16 2006 *)

Crossrefs

Cf. A123883.

Sequence in context: A335535 A372043 A276346 * A256271 A293977 A282649

Adjacent sequences: A123879 A123880 A123881 * A123883 A123884 A123885

Keyword

nonn

Author

Leroy Quet, Oct 16 2006

Extensions

Extended by Ray Chandler, Oct 16 2006

Status

approved