A111170 Semiprimes S such that 3*S + 1 is also a semiprime.

15, 35, 38, 39, 55, 62, 82, 86, 87, 91, 106, 111, 115, 118, 119, 134, 142, 155, 159, 178, 187, 194, 218, 226, 235, 254, 259, 267, 278, 287, 295, 298, 299, 314, 319, 326, 327, 334, 335, 339, 371, 382, 386, 391, 395, 398, 411, 422, 427, 446, 451, 454, 502, 515

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 first kind and Tomaszewski chains of the first 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: 111, 334, 1003; 142, 427, 1282. Length 4: 35, 106, 319, 958; 86, 259, 778, 2335; 187, 562, 1687, 5062.

a(n) is either an even semiprime 2*k where k is a prime such that 6*k+1 is a semiprime, or an odd semiprime 2*k+1 where 3*k+2 is a prime. - Robert Israel, Dec 10 2024

Examples

			n s(n) 3*s + 1
1 15 = 3 * 5 46 = 2 * 23
2 35 = 5 * 7 106 = 2 * 53
3 38 = 2 * 19 115 = 5 * 23
4 39 = 3 * 13 118 = 2 * 59
5 55 = 5 * 11 166 = 2 * 83
6 62 = 2 * 31 187 = 11 * 17
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A111153, A111168, A111171, A111173, A111176.

Programs

Maple

q:= n-> andmap(x-> 2=numtheory[bigomega](x), [n, 3*n+1]):
select(q, [$4..515])[];  # Alois P. Heinz, May 02 2024
# alternative
N:= 10^4: # to get all terms < (N-1)/3
Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):
SP:={seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)}:
sort(convert(SP intersect map(t -> (t-1)/3, SP), list)); # Robert Israel, Dec 10 2024

Mathematica

Select[Range[515],PrimeOmega[#]==2&&PrimeOmega[3*#+1]==2&] (* James C. McMahon, May 01 2024 *)

Formula

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

Extensions

Extended by Ray Chandler, Oct 22 2005