[seqfan] Small perturbations in the natural flow

[Eric Angelini]

Hello SeqFans,

Pick any term 't' of R and jump to the right over 't' terms: 
-- if 't' is odd you'll land on an odd integer; 
-- if 't' is even you'll land on an even integer.

R = 1,2,3,4,6,7,5,8,10,11,9,12,13,15,16,14,18,19,17,20,21,23,
24,22,26,27,25,28,29,31,32,30,33,34,35,36,37,39,40,38,42,43,
41,44,45,47,48,46,49,50,51,52,53,55,56,54,58,59,57,60,61,63,
64,62,65,66,67,68,70,71,73,69,72,74,75,76,77,79,80,78,81,82,
83,84,85,86,87,88,90,91,89,92,93,95,96,94,97,98,99,100,...

R is the lexicographically first such sequence if we want R 
to be a permutation of the Naturals.
Formulas: 
-- if a(n) is odd then a(n+a(n)+1) is odd 
-- if a(n) is even then a(n+a(n)+1) is even
-- always extend R with the smallest available integer not
   yet present in R and not leading to a contradiction.

***************

Replacing "odd" by "prime" in the above definition and
"even" by "composite" gives S and reads (almost):

Pick any term 't' of S and jump to the right over 't' terms: 
-- if 't' is prime you'll land on a prime;
-- if 't' is not a prime you'll land on a composite;
-- always extend S with the smallest available integer not yet
   present in S and not leading to a contradiction.

S = 1,2,4,5,3,6,7,8,11,13,10,9,12,14,17,15,16,18,19,20,23,21,
22,29,24,25,27,26,28,30,31,32,37,33,34,35,36,38,41,39,40,42,
43,44,47,45,46,48,49,50,51,52,53,59,54,55,57,56,58,60,62,63,
61,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,83,...

S is the lexicographically first such sequence if we want S 
to be a permutation of the Naturals.
Formulas: 
-- if a(n) is prime then a(n+a(n)+1) is prime 
-- if a(n) is not prime then a(n+a(n)+1) is not prime
-- always extend S with the smallest available integer not yet
   present and not leading to a contradiction.

Best,
É.