login
A111153
Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.
27
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
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