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A307121
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Number of Lucasian primes less than 10^n.
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1
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1, 4, 19, 100, 581, 3912, 28091, 211700, 1655601, 13286320, 109058381, 911436949, 7731247492
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OFFSET
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1,2
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COMMENTS
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The Lucasian primes are those Sophie Germain primes of the form 4k + 3. They are interesting because if they are infinite in number, then the sequence of Mersenne numbers with prime exponents contains an infinite number of composite integers.
Conjecture: about half of all Sophie Germain primes are Lucasian primes, and the rest are either 2 or a prime of the form 4k + 1.
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LINKS
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EXAMPLE
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There are 4 Lucasian primes below 10^2: {3,11,23,83}, therefore a(2) = 4.
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MATHEMATICA
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c = 0; r = 10; s = {}; Do[If[p > r, AppendTo[s, c]; r *= 10]; If[PrimeQ[p] && PrimeQ[2p + 1], c++], {p, 3, 1000003, 4}]; s (* Amiram Eldar, Mar 27 2019 *)
lucSophies = Select[4Range[2500000] - 1, PrimeQ[#] && PrimeQ[2# + 1] &]; Table[Length[Select[lucSophies, # < 10^n &]], {n, 0, 7}]
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PROG
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(PARI) a(n) = { my(t=0); forprime(p=2, 10^n, p%4==3 && ispseudoprime(1+(2*p)) && t++); t } \\ Dana Jacobsen, Mar 28 2019
(Perl) use ntheory ":all"; sub a { my($n, $t)=(shift, 0); forprimes { $t++ if ($_&3) == 3 && is_prime(1+($_<<1)) } 10**$n; $t; } # Dana Jacobsen, Mar 28 2019
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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