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A262350
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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.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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: 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
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MAPLE
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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);
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CROSSREFS
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Primes whose binary expansion begins with binary expansion of 1, 2, 3, 4, 5, 6, 7: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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