A002515 Lucasian primes: p == 3 (mod 4) with 2*p+1 prime.

3, 11, 23, 83, 131, 179, 191, 239, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2339, 2351, 2399, 2459, 2543, 2699, 2819, 2903, 2939, 2963, 3023, 3299

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

2*p+1 divides A000225(p), the p-th Mersenne number. - Lekraj Beedassy, Jun 23 2003

Also primes p such that 2^(2*p+1) - 1 divides 2^(2^p-1) - 1. - Arkadiusz Wesolowski, May 26 2011

Intersection of A005384 (Sophie Germain primes) and A002145. - Jeppe Stig Nielsen, Aug 03 2020

References

• A. J. C. Cunningham, On Mersenne's numbers, Reports of the British Association for the Advancement of Science, 1894, pp. 563-564.
• L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 27.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 76.
• Daniel Shanks, "Solved and Unsolved Problems in Number Theory," Fourth Edition, Chelsea Publishing Co., NY, 1993, page 28.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from T. D. Noe)

Crossrefs

Intersection of A002145 and A005384.

Programs

MATLAB

p=primes(1500); m=1;
for u=1:length(p)
   if and(isprime(2*p(u)+1)==1, mod(p(u),4)==3) ; sol(m)=p(u); m=m+1;  end;
end
sol % Marius A. Burtea, Mar 26 2019

Magma

[p: p in PrimesUpTo(6000) | IsPrime(2*p+1) and p mod 4 in [3]]; // Vincenzo Librandi, Sep 03 2016

Mathematica

Select[Range[10^4], Mod[ #, 4] == 3 && PrimeQ[ # ] && PrimeQ[2# + 1] & ]
Select[Prime[Range[500]],Mod[#,4]==3&&PrimeQ[2#+1]&] (* Harvey P. Dale, Mar 15 2016 *)

PARI

is(n)=n%4>2 && isprime(n) && isprime(2*n+1) \\ Charles R Greathouse IV, Apr 01 2013

PARI

list(lim)=my(v=List()); forprimestep(p=3,lim\1,4, if(isprime(2*p+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 25 2024

Formula

a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

Extensions

More terms from Robert G. Wilson v, Mar 07 2002