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A299704 List of primes prime(r) such that prime(r)-prime(r-1)=30, prime(r-1)-prime(r-2)=8 and prime(r-2)-prime(r-3)=6.

%I A299704 #31 May 13 2020 02:27:31
%S A299704 4327,91621,111697,123001,190027,240997,243517,244291,300277,309667,
%T A299704 315937,317827,362137,393517,440131,457087,467587,517861,554167,
%U A299704 567097,590071,609571,617917,640771,651727,653311,719101,776551,788071,793591,804157,809491,812431,850177,861391,1007857,1070287
%N A299704 List of primes prime(r) such that prime(r)-prime(r-1)=30, prime(r-1)-prime(r-2)=8 and prime(r-2)-prime(r-3)=6.
%C A299704 These are the primes of a056240-type 3(30,8,6); k=3 (see definition in A293652).
%C A299704 A prime of a056240-type 3 is a prime, prime(r)>3, such that prime(r-3) is the greatest prime factor of the smallest composite number whose prime divisors (with multiplicity) sum to prime(r).
%C A299704 Conjecture: Sequence has infinitely many terms.
%C A299704 Note: p~3(30,8,6) is one particular form of a prime of a056240-type 3; there are others, e.g., 3(30,12,2), 3(24,6,2), 3(36,6,4), 3(38,10,2), etc. All such prime sequences are also conjectured to produce infinitely many terms.
%C A299704 All terms == 1 (mod 3). - _Robert Israel_, May 13 2020
%H A299704 Robert Israel, <a href="/A299704/b299704.txt">Table of n, a(n) for n = 1..2000</a>
%F A299704 For every prime(r) in this sequence A288814(prime(r)) = prime(r-3)*A056240(prime(r) - prime(r-3)) = prime(r-3)*A288814(prime(r) - prime(r-3)).
%e A299704 a(1)=4327=prime(591), the first prime of a056240-type 3. Prime(590)=4297, prime(589)=4289, prime(588)=4283. 4327-4297=30, 4297-4289=8, 4289-4283=6.
%p A299704 N:=2000000:
%p A299704 for X from 100 to N do
%p A299704 if isprime(X) then
%p A299704 A:=prevprime(X);
%p A299704 B:=prevprime(A);
%p A299704 C:=prevprime(B);
%p A299704 a:=X-A;
%p A299704 b:=A-B;
%p A299704 c:=B-C;
%p A299704 if a=30 and b=8 and c=6 then print(X);
%p A299704 end if
%p A299704 end if
%p A299704 end if
%p A299704 end do
%t A299704 With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Select[s, Differences@ # == {6, 8, 30} &][[All, -1]]] (* _Michael De Vlieger_, Feb 18 2018 *)
%Y A299704 Cf. A056240, A288814, A293652, A295185, A299110.
%K A299704 nonn
%O A299704 1,1
%A A299704 _David James Sycamore_, Feb 17 2018