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

1,1

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A381556 A253055 A338063 * A134335 A257591 A284406

Adjacent sequences: A111167 A111168 A111169 * A111171 A111172 A111173

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 21 2005

Extensions

Extended by Ray Chandler, Oct 22 2005

Status

approved