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A308334 Lexicographically earliest sequence of distinct positive numbers such that for any n > 0, a(n) OR a(n+1) is a prime number (where OR denotes the bitwise OR operator).

%I A308334 #21 Sep 10 2022 08:38:46
%S A308334 1,2,3,4,5,6,7,16,13,8,11,9,10,21,12,17,14,19,15,18,23,20,25,22,27,28,
%T A308334 29,24,31,26,33,36,37,32,41,34,43,35,40,39,42,45,38,47,44,49,52,53,48,
%U A308334 59,50,57,51,56,61,60,67,62,65,63,64,71,58,69,66,77,54
%N A308334 Lexicographically earliest sequence of distinct positive numbers such that for any n > 0, a(n) OR a(n+1) is a prime number (where OR denotes the bitwise OR operator).
%C A308334 By Dirichlet's theorem on arithmetic progressions, we can always extend the sequence: say a(n) < 2^k, then a(n) OR 1 and 2^k are coprime and there are infinitely many prime numbers of the form (a(n) OR 1) + m*2^k = a(n) OR (1 + m*2^k) and we can extend the sequence.
%C A308334 Will every integer appear in this sequence?
%C A308334 Numerous sequences are based on the same model: the sequence is the lexicographically earliest sequence of distinct positive terms such that some function in two variables yields prime numbers when applied to consecutive terms:
%C A308334   f(u,v)      Analog sequence
%C A308334   -------     -----------------
%C A308334   u OR v      a (this sequence)
%C A308334   u + v       A055265
%C A308334   u*v + 1     A073666
%C A308334   u*v - 1     A081943
%C A308334   abs(u-v)    A065186
%C A308334   max(u,v)    A282649
%C A308334   u^2 + v^2   A100208
%C A308334 The appearance of numbers much earlier or later than their corresponding index is flagged strikingly in the plot2 graph of a(n)/n (see links). - _Peter Munn_, Sep 10 2022
%H A308334 Rémy Sigrist, <a href="/A308334/b308334.txt">Table of n, a(n) for n = 1..10000</a>
%H A308334 Peter Munn, <a href="https://oeis.org/plot2a?name1=A308334&amp;name2=A000027&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Plot2 graph of a(n)/n</a>
%e A308334 The first terms, alongside a(n) OR a(n+1), are:
%e A308334   n   a(n)  a(n) OR a(n+1)
%e A308334   --  ----  --------------
%e A308334    1     1               3
%e A308334    2     2               3
%e A308334    3     3               7
%e A308334    4     4               5
%e A308334    5     5               7
%e A308334    6     6               7
%e A308334    7     7              23
%e A308334    8    16              29
%e A308334    9    13              13
%e A308334   10     8              11
%e A308334   11    11              11
%e A308334   12     9              11
%o A308334 (PARI) s=0; v=1; for (n=1, 67, s+=2^v; print1 (v ", "); for (w=1, oo, if (!bittest(s,w) && isprime(o=bitor(v,w)), v=w; break)))
%o A308334 (Python)
%o A308334 from sympy import isprime
%o A308334 from itertools import count, islice
%o A308334 def agen():
%o A308334     aset, k, mink = {1}, 1, 2
%o A308334     for n in count(1):
%o A308334         an = k; yield an; aset.add(an)
%o A308334         s, k = set(str(an)), mink
%o A308334         while k in aset or not isprime(an|k): k += 1
%o A308334         while mink in aset: mink += 1
%o A308334 print(list(islice(agen(), 67))) # _Michael S. Branicky_, Sep 10 2022
%Y A308334 See A308340 for the corresponding prime numbers.
%Y A308334 See A055265, A065186, A073666, A081943, A100208, A282649 for similar sequences.
%K A308334 nonn,base
%O A308334 1,2
%A A308334 _Rémy Sigrist_, May 20 2019