OFFSET
1,2
COMMENTS
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.
LINKS
Simon Davis, Arithmetical sequences for the exponents of composite Mersenne numbers, Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 19-26.
EXAMPLE
There are 4 Lucasian primes below 10^2: {3,11,23,83}, therefore a(2) = 4.
MATHEMATICA
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Rodolfo Ruiz-Huidobro, Mar 26 2019
EXTENSIONS
a(9)-a(11) from Amiram Eldar, Mar 27 2019
a(12) from Amiram Eldar, Mar 31 2019
a(13) from Dana Jacobsen, Apr 02 2019
STATUS
approved