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A299110 Prime(r) for r such that prime(r) - prime(r-1) = 12 and prime(r-1) - prime(r-2) = 2.

%I A299110 #39 Feb 29 2020 12:50:54
%S A299110 211,631,673,1801,3181,3271,3343,3571,3943,4561,4813,5431,6673,6883,
%T A299110 7321,7573,7603,7963,8443,8641,9643,9733,9781,9871,10513,10723,10903,
%U A299110 11083,11131,11731,11953,12391,13411,14401,14461,15373,15661,15901,16843,17203,17431,17761,17851,17971,18301,18553,20161,20521,20563,20731
%N A299110 Prime(r) for r such that prime(r) - prime(r-1) = 12 and prime(r-1) - prime(r-2) = 2.
%C A299110 These are the primes of a056240-type 2(12,2); k=2 (see definition in A293652). prime(r-2) is the greatest prime factor of the smallest composite number whose prime divisors (with multiplicity) sum to prime(r).
%C A299110 Conjecture: Sequence has infinitely many terms. Note: p~2(12,2) is just one particular form of a prime of A056240-type k=2; there are others, e.g., 2(18,2), 2(18,4), 2(28,12), 2(24,10). All such prime sequences are also conjectured to produce infinitely many terms.
%H A299110 Robert Israel, <a href="/A299110/b299110.txt">Table of n, a(n) for n = 1..10000</a>
%F A299110 For every prime(r) in this sequence A288814(prime(r)) = prime(r-2)*A056240(prime(r) - prime(r-2)) = prime(r-2)*A288814(prime(r) - prime(r-2)).
%e A299110 a(1)=211=prime(47), the first prime of type k=2. prime(46)=199 and prime(45)=197; 211-199=12 and 199-197=2.
%p A299110 N:=21000:
%p A299110 for X from 2 to N do
%p A299110 if isprime(X) then
%p A299110 A:=prevprime(X);
%p A299110 B:=prevprime(A);
%p A299110 a:=X-A;
%p A299110 b:=A-B;
%p A299110 if a=12 and b=2 then print(X);
%p A299110 end if
%p A299110 end if
%p A299110 end if
%p A299110 end do
%p A299110 # alternative:
%p A299110 P:= select(isprime, {seq(i,i=3..10^6,2)}):
%p A299110 Q:= P intersect map(t -> t-12, P) intersect map(t -> t+2, P):
%p A299110 Q:= remove(t -> ormap(isprime, [seq(t+i,i=2..10,2)]), Q):
%p A299110 map(t -> t+12, Q); # _Robert Israel_, Feb 16 2018
%t A299110 Select[Partition[Prime[Range[2500]],3,1],Differences[#]=={2,12}&][[All,3]] (* _Harvey P. Dale_, Feb 29 2020 *)
%o A299110 (PARI) isok(p) =  isprime(p) && (pp=precprime(p-1)) && (p-pp == 12) && (ppp=precprime(pp-1)) && (pp-ppp == 2); \\ _Michel Marcus_, Feb 16 2018
%Y A299110 Cf. A056240, A288814, A293652, A295185.
%K A299110 nonn
%O A299110 1,1
%A A299110 _David James Sycamore_, Feb 16 2018