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%I #21 Sep 10 2022 08:38:46
%S 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 29,24,31,26,33,36,37,32,41,34,43,35,40,39,42,45,38,47,44,49,52,53,48,
%U 59,50,57,51,56,61,60,67,62,65,63,64,71,58,69,66,77,54
%N 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 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 Will every integer appear in this sequence?
%C 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 f(u,v) Analog sequence
%C ------- -----------------
%C u OR v a (this sequence)
%C u + v A055265
%C u*v + 1 A073666
%C u*v - 1 A081943
%C abs(u-v) A065186
%C max(u,v) A282649
%C u^2 + v^2 A100208
%C 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 Rémy Sigrist, <a href="/A308334/b308334.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Munn, <a href="https://oeis.org/plot2a?name1=A308334&name2=A000027&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawlines=true">Plot2 graph of a(n)/n</a>
%e The first terms, alongside a(n) OR a(n+1), are:
%e n a(n) a(n) OR a(n+1)
%e -- ---- --------------
%e 1 1 3
%e 2 2 3
%e 3 3 7
%e 4 4 5
%e 5 5 7
%e 6 6 7
%e 7 7 23
%e 8 16 29
%e 9 13 13
%e 10 8 11
%e 11 11 11
%e 12 9 11
%o (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 (Python)
%o from sympy import isprime
%o from itertools import count, islice
%o def agen():
%o aset, k, mink = {1}, 1, 2
%o for n in count(1):
%o an = k; yield an; aset.add(an)
%o s, k = set(str(an)), mink
%o while k in aset or not isprime(an|k): k += 1
%o while mink in aset: mink += 1
%o print(list(islice(agen(), 67))) # _Michael S. Branicky_, Sep 10 2022
%Y See A308340 for the corresponding prime numbers.
%Y See A055265, A065186, A073666, A081943, A100208, A282649 for similar sequences.
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, May 20 2019
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