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A111170
Semiprimes S such that 3*S + 1 is also a semiprime.
7
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
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 21 2005
EXTENSIONS
Extended by Ray Chandler, Oct 22 2005
STATUS
approved