A111173 - OEIS

A111173

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

%I #30 Jun 22 2022 09:26:08

%S 52,76,130,171,172,212,238,318,322,325,332,357,370,387,388,402,423,

%T 430,436,442,465,507,508,556,604,610,654,665,670,710,722,747,759,762,

%U 772,775,786,790,805,814,822,826,847,874,885,902,906,916,927,942,987,1004

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

%C 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 Zak Seidov, <a href="/A111173/b111173.txt">Table of n, a(n) for n = 1..1000</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Triprimes">Triprimes</a>

%F {a(n)} = a(n) is an element of A014612 and 2*a(n)+1 is an element of A014612.

%e n k = a(n) 2k + 1

%e = ================ ================

%e 1 52 = 2^2 * 13 105 = 3 * 5 * 7

%e 2 76 = 2^2 * 19 153 = 3^2 * 17

%e 3 130 = 2 * 5 * 13 261 = 3^2 * 29

%e 4 171 = 3^2 * 19 343 = 7^3

%e 5 172 = 2^2 * 43 345 = 3 * 5 * 23

%e 6 212 = 2^2 * 53 425 = 5^2 * 17

%t fQ[n_]:=PrimeOmega[n] == 3 == PrimeOmega[2 n + 1]; Select[Range@1100, fQ] (* _Vincenzo Librandi_, Aug 19 2018 *)

%o (PARI) is(n)=bigomega(n)==3 && bigomega(2*n+1)==3 \\ _Charles R Greathouse IV_, Feb 01 2017

%o (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 Cf. A005384, A014612, A111153, A111168, A111170, A111171, A111176.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Oct 21 2005

%E Extended by _Ray Chandler_, Oct 22 2005

%E Edited by _Jon E. Schoenfield_, Aug 18 2018