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A111173 Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.

%I A111173 #30 Jun 22 2022 09:26:08
%S A111173 52,76,130,171,172,212,238,318,322,325,332,357,370,387,388,402,423,
%T A111173 430,436,442,465,507,508,556,604,610,654,665,670,710,722,747,759,762,
%U A111173 772,775,786,790,805,814,822,826,847,874,885,902,906,916,927,942,987,1004
%N A111173 Sophie Germain triprimes: k and 2k + 1 are both the product of 3 primes, not necessarily distinct.
%C A111173 There should also be triprime chains of length j analogous to Cunningham chains of the first kind and Tomaszewski chains of the first kind. A triprime chain of length j is a sequence of triprimes a(1) < a(2) < ... < a(j) such that a(i+1) = 2*a(i) + 1 for i = 1, ..., j-1. The first of these are: Length 3: 332, 665, 1331 = 11^3; 387, 775, 1551 = 3 * 11 * 47.
%H A111173 Zak Seidov, <a href="/A111173/b111173.txt">Table of n, a(n) for n = 1..1000</a>
%H A111173 OEIS Wiki, <a href="http://oeis.org/wiki/Triprimes">Triprimes</a>
%F A111173 {a(n)} = a(n) is an element of A014612 and 2*a(n)+1 is an element of A014612.
%e A111173 n      k = a(n)           2k + 1
%e A111173 =  ================  ================
%e A111173 1   52 = 2^2 * 13    105 = 3 * 5 * 7
%e A111173 2   76 = 2^2 * 19    153 = 3^2 * 17
%e A111173 3  130 = 2 * 5 * 13  261 = 3^2 * 29
%e A111173 4  171 = 3^2 * 19    343 = 7^3
%e A111173 5  172 = 2^2 * 43    345 = 3 * 5 * 23
%e A111173 6  212 = 2^2 * 53    425 = 5^2 * 17
%t A111173 fQ[n_]:=PrimeOmega[n] == 3 == PrimeOmega[2 n + 1]; Select[Range@1100, fQ] (* _Vincenzo Librandi_, Aug 19 2018 *)
%o A111173 (PARI) is(n)=bigomega(n)==3 && bigomega(2*n+1)==3 \\ _Charles R Greathouse IV_, Feb 01 2017
%o A111173 (Magma) Is3primes:=func<i|&+[d[2]: d in Factorization(i)] eq 3>; [n: n in [2..1200] | Is3primes(n) and Is3primes(2*n+1)]; // _Vincenzo Librandi_, Aug 19 2018
%Y A111173 Cf. A005384, A014612, A111153, A111168, A111170, A111171, A111176.
%K A111173 easy,nonn
%O A111173 1,1
%A A111173 _Jonathan Vos Post_, Oct 21 2005
%E A111173 Extended by _Ray Chandler_, Oct 22 2005
%E A111173 Edited by _Jon E. Schoenfield_, Aug 18 2018