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A355166 Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.

%I A355166 #8 Jun 23 2022 08:11:36
%S A355166 2,1,4,3,8,17,16,5,20,21,32,19,18,33,64,7,6,13,12,9,10,41,40,35,34,37,
%T A355166 68,65,66,97,96,11,14,25,24,67,26,73,80,23,22,85,84,81,82,129,128,71,
%U A355166 72,69,76,75,74,137,136,131,70,133,132,193,130,257,256,15,28
%N A355166 Lexicographically earliest sequence of distinct positive integers such for any n > 0, n and a(n) are coprime and have no common 1-bits in their binary expansions.
%C A355166 This sequence combines features of A065190 and of A238757.
%C A355166 This sequence is a self-inverse permutation of the nonnegative integers, without fixed points.
%C A355166 This sequence is well defined:
%C A355166 - if n is odd, then we can extend the sequence with a power of 2 > n,
%C A355166 - if n < 2^k is even, then we can extend the sequence with a prime number of the form 1 + t*2^k (Dirichlet's theorem on arithmetic progressions guarantees us that there is an infinity of such prime numbers).
%C A355166 When n is odd, a(n) is even and vice-versa.
%H A355166 Rémy Sigrist, <a href="/A355166/a355166.gp.txt">PARI program</a>
%H A355166 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A355166 The first terms, alongside binary expansions and distinct prime factors, are:
%e A355166   n   a(n)  bin(n)  bin(a(n))  dpf(n)  dpf(a(n))
%e A355166   --  ----  ------  ---------  ------  ---------
%e A355166    1     2       1         10  {}      {2}
%e A355166    2     1      10          1  {2}     {}
%e A355166    3     4      11        100  {3}     {2}
%e A355166    4     3     100         11  {2}     {3}
%e A355166    5     8     101       1000  {5}     {2}
%e A355166    6    17     110      10001  {2, 3}  {17}
%e A355166    7    16     111      10000  {7}     {2}
%e A355166    8     5    1000        101  {2}     {5}
%e A355166    9    20    1001      10100  {3}     {2, 5}
%e A355166   10    21    1010      10101  {2, 5}  {3, 7}
%o A355166 (PARI) See Links section.
%o A355166 (Python)
%o A355166 from math import gcd
%o A355166 from itertools import count, islice
%o A355166 def agen(): # generator of terms
%o A355166     aset, mink = set(), 1
%o A355166     for n in count(1):
%o A355166         an = mink
%o A355166         while an in aset or n&an or gcd(n, an)!=1: an += 1
%o A355166         aset.add(an); yield an
%o A355166         while mink in aset: mink += 1
%o A355166 print(list(islice(agen(), 65))) # _Michael S. Branicky_, Jun 22 2022
%Y A355166 Cf. A065190, A238757, A352633.
%K A355166 nonn,base
%O A355166 1,1
%A A355166 _Rémy Sigrist_, Jun 22 2022