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

52, 76, 130, 171, 172, 212, 238, 318, 322, 325, 332, 357, 370, 387, 388, 402, 423, 430, 436, 442, 465, 507, 508, 556, 604, 610, 654, 665, 670, 710, 722, 747, 759, 762, 772, 775, 786, 790, 805, 814, 822, 826, 847, 874, 885, 902, 906, 916, 927, 942, 987, 1004

Offset: 1

Author

Jonathan Vos Post, Oct 21 2005

Keywords

• easy
• nonn

Comments

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.

Examples

			n      k = a(n)           2k + 1
=  ================  ================
1   52 = 2^2 * 13    105 = 3 * 5 * 7
2   76 = 2^2 * 19    153 = 3^2 * 17
3  130 = 2 * 5 * 13  261 = 3^2 * 29
4  171 = 3^2 * 19    343 = 7^3
5  172 = 2^2 * 43    345 = 3 * 5 * 23
6  212 = 2^2 * 53    425 = 5^2 * 17
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000
• OEIS Wiki, Triprimes

Crossrefs

Cf. A005384, A014612, A111153, A111168, A111170, A111171, A111176.

Programs

Magma

Is3primes:=func; [n: n in [2..1200] | Is3primes(n) and Is3primes(2*n+1)]; // Vincenzo Librandi, Aug 19 2018

Mathematica

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

PARI

is(n)=bigomega(n)==3 && bigomega(2*n+1)==3 \\ Charles R Greathouse IV, Feb 01 2017

Formula

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

Extensions

Extended by Ray Chandler, Oct 22 2005

Edited by Jon E. Schoenfield, Aug 18 2018