A111171 Semiprimes S such that 3*S - 1 is also a semiprime.

9, 21, 22, 25, 26, 49, 62, 65, 69, 74, 85, 93, 121, 122, 129, 133, 141, 146, 158, 161, 166, 178, 185, 194, 205, 209, 221, 249, 253, 262, 265, 289, 298, 302, 305, 309, 346, 358, 361, 365, 381, 382, 386, 413, 446, 466, 473, 485, 489, 493, 501, 505, 514, 526, 553

Offset: 1

Author

Jonathan Vos Post, Oct 21 2005

Keywords

• easy
• nonn

Comments

This is analogous to Sophie Germain semiprimes A111153 and the chains shown are analogous to Cunningham chains of the second kind and Tomaszewski chains of the second kind. Define a 3n-1 semiprime chain of length k. This is a sequence of semiprimes s(1) < s(2) < ... < s(k) such that s(i+1) = 3*s(i) - 1 for i = 1, ..., k-1. Length 3: 9, 26, 77; 49, 146, 437; 65, 194, 581; 129, 386, 1157; 158, 473, 1418; 187, 562, 1685. Length 4: 74, 221, 662, 1985; 122, 365, 1094, 3281. Length 5: 21, 62, 185, 554, 1661.

Examples

			n s(n) 3 *s -1
1 9 = 3^2 26 = 2 * 13
2 21 = 3 * 7 62 = 2 * 31
3 22 = 2 * 11 65 = 5 * 13
4 25 = 5^2 74 = 2 * 37
5 26 = 2 * 13 77 = 7 * 11
6 49 = 7^2 146 = 2 * 73
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358, A111153, A111168, A111170, A111173, A111176.

Programs

Mathematica

Select[Range[600],PrimeOmega[#]==PrimeOmega[3#-1]==2&] (* Harvey P. Dale, Jun 20 2018 *)

Formula

{a(n)} = a(n) is an element of A001358 and 3*a(n)-1 is an element of A001358.

Extensions

Corrected and extended by Ray Chandler, Oct 22 2005