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A111153
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Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.
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26
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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
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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a(4)=34 because 34 is the 4th semiprime such that 2*34+1=69 is also a semiprime.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 19 2005
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EXTENSIONS
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STATUS
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approved
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