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

1,1

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.

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A365861 A107961 A051864 * A265438 A145368 A266826

Adjacent sequences: A111150 A111151 A111152 * A111154 A111155 A111156

Keyword

nonn

Author

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

Extensions

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

Status

approved