A307121 Number of Lucasian primes less than 10^n.

1, 4, 19, 100, 581, 3912, 28091, 211700, 1655601, 13286320, 109058381, 911436949, 7731247492

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

Table of n, a(n) for n=1..13.

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

Cf. A092816, A002515, A307176.

Sequence in context: A218185 A369672 A086624 * A261490 A277956 A307678

Adjacent sequences: A307118 A307119 A307120 * A307122 A307123 A307124

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