A111153 Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.

4, 10, 25, 34, 38, 46, 55, 57, 77, 91, 93, 106, 118, 123, 129, 133, 143, 145, 159, 161, 169, 177, 185, 201, 203, 205, 206, 213, 218, 226, 235, 259, 267, 289, 291, 295, 298, 305, 314, 327, 334, 335, 339, 358, 361, 365, 377, 381, 394, 395, 403, 407, 415, 417

Offset: 1

Author

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 19 2005

Keywords

• nonn

Comments

Define a generalized Sophie Germain n-prime of degree m, p, to be an n-prime (n-almost prime) such that 2p+1 is an m-prime (m-almost prime). For example, p=24 is a Sophie Germain 4-prime of degree 2 because 24 is a 4-prime and 2*24+1=49 is a 2-prime. Then this sequence gives all the Sophie Germain 2-primes of degree 2.

Examples

			a(4)=34 because 34 is the 4th semiprime such that 2*34+1=69 is also a semiprime.
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..7675 (first 1000 terms from T. D. Noe)

Crossrefs

Cf. A005384, A001358, A111168, A111170, A111171, A111173, A111176, A176896.

Programs

Magma

f:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..500] | f(n) and f(2*n+1)]; // Marius A. Burtea, Jan 04 2019

Mathematica

SemiPrimeQ[n_] := (Plus@@Transpose[FactorInteger[n]][[2]]==2); Select[Range[2, 500], SemiPrimeQ[ # ]&&SemiPrimeQ[2#+1]&] (* T. D. Noe, Oct 20 2005 *)
fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[445], fQ[ # ] && fQ[2# + 1] &] (* Robert G. Wilson v, Oct 20 2005 *)
Flatten@Position[PrimeOmega@{#,1+2*#}&/@Range@1000,{2,2}] (* Hans Rudolf Widmer, Nov 25 2023 *)

PARI

isok(n) = (bigomega(n) == 2) && (bigomega(2*n+1) == 2); \\ Michel Marcus, Jan 04 2019

Formula

a(n) = (A176896(n) - 1)/2. - Zak Seidov, Sep 10 2012

Extensions

Corrected and extended by T. D. Noe, Ray Chandler and Robert G. Wilson v, Oct 20 2005