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A262350 a(1) = 2. For n>1, let s denote the binary string of a(n-1) with the leftmost 1 and following consecutive 0's removed. Then a(n) is the smallest prime not yet present whose binary representation begins with s. 7
2, 3, 5, 7, 13, 11, 29, 53, 43, 23, 31, 61, 59, 109, 181, 107, 173, 367, 223, 191, 127, 509, 1013, 4013, 3931, 3767, 13757, 11131, 2939, 1783, 3037, 1979, 3821, 3547, 1499, 1901, 877, 2927, 1759, 1471, 1789, 1531, 2029, 2011, 7901, 60887, 56239, 93887, 28351 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This sequence is infinite.  The number of primes that are not in this sequence is conjectured to be infinite.

Proof of first statement, following a comment from David W. Wilson: It follows from standard results about primes in short intervals (see for example Harman, 1982) that there are infinitely many numbers in any base b starting with any nonzero prefix c. So there are infinitely many primes whose binary expansion begins with s, and so a(n) always exists. - N. J. A. Sloane, Sep 19 2015

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..589

G. Harman, Primes in short intervals, Math. Zeit., 180 (1982), 335-348.

EXAMPLE

: 10                             ... 2

:   11                           ... 3

:    101                         ... 5

:      111                       ... 7

:       1101                     ... 13

:        1011                    ... 11

:          11101                 ... 29

:           110101               ... 53

:            101011              ... 43

:              10111             ... 23

:                11111           ... 31

:                 111101         ... 61

:                  111011        ... 59

:                   1101101      ... 109

:                    10110101    ... 181

:                      1101011   ... 107

:                       10101101 ... 173

MAPLE

b:= proc() true end:

a:= proc(n) option remember; local h, k, ok, p, t;

      if n=1 then p:=2

    else h:= (k-> irem(k, 2^(ilog2(k))))(a(n-1)); p:= h;

         ok:= isprime(p) and b(p);

         for t while not ok do

           for k to 2^t-1 while not ok do p:= h*2^t+k;

             ok:= isprime(p) and b(p)

           od

         od

      fi; b(p):= false; p

    end:

seq(a(n), n=1..70);

CROSSREFS

Binary analog of A262283.

Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.

Cf. A262365.

Sequence in context: A316885 A225039 A264731 * A228891 A168484 A217907

Adjacent sequences:  A262347 A262348 A262349 * A262351 A262352 A262353

KEYWORD

nonn,base

AUTHOR

Alois P. Heinz, Sep 18 2015

STATUS

approved

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Last modified July 18 11:48 EDT 2019. Contains 325139 sequences. (Running on oeis4.)