

A111153


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


25



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
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OFFSET

1,1


COMMENTS

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


MAPLE

with(numtheory): P:=proc(n) if bigomega(n)=2 and bigomega(2*n+1)=2
then n; fi; end: seq(P(i), i=1..10^4); # Paolo P. Lava, Mar 10 2017


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 *)


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: A127070 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



