%I #19 Dec 12 2024 09:27:50
%S 15,35,38,39,55,62,82,86,87,91,106,111,115,118,119,134,142,155,159,
%T 178,187,194,218,226,235,254,259,267,278,287,295,298,299,314,319,326,
%U 327,334,335,339,371,382,386,391,395,398,411,422,427,446,451,454,502,515
%N Semiprimes S such that 3*S + 1 is also a semiprime.
%C 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.
%C 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
%H Robert Israel, <a href="/A111170/b111170.txt">Table of n, a(n) for n = 1..10000</a>
%F {a(n)} = a(n) is an element of A001358 and 3*a(n)+1 is an element of A001358.
%e n s(n) 3*s + 1
%e 1 15 = 3 * 5 46 = 2 * 23
%e 2 35 = 5 * 7 106 = 2 * 53
%e 3 38 = 2 * 19 115 = 5 * 23
%e 4 39 = 3 * 13 118 = 2 * 59
%e 5 55 = 5 * 11 166 = 2 * 83
%e 6 62 = 2 * 31 187 = 11 * 17
%p q:= n-> andmap(x-> 2=numtheory[bigomega](x), [n, 3*n+1]):
%p select(q, [$4..515])[]; # _Alois P. Heinz_, May 02 2024
%p # alternative
%p N:= 10^4: # to get all terms < (N-1)/3
%p Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):
%p SP:={seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)}:
%p sort(convert(SP intersect map(t -> (t-1)/3, SP), list)); # _Robert Israel_, Dec 10 2024
%t Select[Range[515],PrimeOmega[#]==2&&PrimeOmega[3*#+1]==2&] (* _James C. McMahon_, May 01 2024 *)
%Y Cf. A001358, A111153, A111168, A111171, A111173, A111176.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Oct 21 2005
%E Extended by _Ray Chandler_, Oct 22 2005