A308340 -id:A308340 - cp's OEIS frontend

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).

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, 29, 24, 31, 26, 33, 36, 37, 32, 41, 34, 43, 35, 40, 39, 42, 45, 38, 47, 44, 49, 52, 53, 48, 59, 50, 57, 51, 56, 61, 60, 67, 62, 65, 63, 64, 71, 58, 69, 66, 77, 54

Offset: 1

Rémy Sigrist, May 20 2019

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.
Will every integer appear in this sequence?
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:
  f(u,v)      Analog sequence
  -------     -----------------
  u OR v      a (this sequence)
  u + v       A055265
  u*v + 1     A073666
  u*v - 1     A081943
  abs(u-v)    A065186
  max(u,v)    A282649
  u^2 + v^2   A100208
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

			The first terms, alongside a(n) OR a(n+1), are:
  n   a(n)  a(n) OR a(n+1)
  --  ----  --------------
   1     1               3
   2     2               3
   3     3               7
   4     4               5
   5     5               7
   6     6               7
   7     7              23
   8    16              29
   9    13              13
  10     8              11
  11    11              11
  12     9              11

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Peter Munn, Plot2 graph of a(n)/n

See A308340 for the corresponding prime numbers.

See A055265, A065186, A073666, A081943, A100208, A282649 for similar sequences.

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)))

from sympy import isprime
from itertools import count, islice
def agen():
    aset, k, mink = {1}, 1, 2
    for n in count(1):
        an = k; yield an; aset.add(an)
        s, k = set(str(an)), mink
        while k in aset or not isprime(an|k): k += 1
        while mink in aset: mink += 1
print(list(islice(agen(), 67))) # Michael S. Branicky, Sep 10 2022

A337013 Prime numbers arising in A336817.

Original entry on oeis.org

3, 7, 3, 5, 7, 3, 13, 2, 3, 2, 5, 2, 3, 2, 31, 2, 3, 2, 7, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 61, 2, 3, 2, 5, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 31, 2, 3, 2, 7, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 127, 2, 3, 2, 7, 2, 3, 2, 13, 2, 3, 2, 5, 2, 3, 2, 31, 2, 3, 2, 7

Offset: 1

Rémy Sigrist, Nov 21 2020

This sequence contains large runs of consecutive terms that are repeated later on.

			a(7) = A336817(7) XOR A336817(8) = 7 XOR 10 = 13.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A308340, A336817.

s=0; v=1; for (n=1, 83, s+=2^v; for (w=1, oo, if (!bittest(s, w) && isprime(p=bitxor(v, w)), print1 (p ", "); v=w; break)))

a(n) = A336817(n) XOR A336817(n+1) (where XOR denotes the bitwise XOR operator).

A352723 a(n) = A352722(n) AND A352722(n+1) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 2, 2, 2, 5, 5, 13, 11, 2, 2, 2, 17, 17, 17, 17, 19, 2, 2, 2, 2, 2, 2, 2, 2, 3, 29, 5, 37, 37, 41, 41, 43, 37, 53, 53, 61, 59, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 17, 17, 17, 17, 5, 5, 5, 5, 73, 73, 71, 83, 89, 89, 73, 73, 67, 97, 97, 101, 101, 109

Offset: 1

Rémy Sigrist, Mar 30 2022

All terms are prime.

			a(5) = A352722(5) AND A352722(6) = 7 AND 5 = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A308340, A337013, A352722.

s=0; v=2; for (n=1, 74, s+=2^v; for (w=1, oo, if (!bittest(s, w) && isprime(p=bitand(v, w)), v=w; print1(p", "); break)))

cp's OEIS Frontend

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).

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A337013 Prime numbers arising in A336817.

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A352723 a(n) = A352722(n) AND A352722(n+1) (where AND denotes the bitwise AND operator).

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