%I #31 May 13 2020 02:27:31

%S 4327,91621,111697,123001,190027,240997,243517,244291,300277,309667,

%T 315937,317827,362137,393517,440131,457087,467587,517861,554167,

%U 567097,590071,609571,617917,640771,651727,653311,719101,776551,788071,793591,804157,809491,812431,850177,861391,1007857,1070287

%N 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 These are the primes of a056240-type 3(30,8,6); k=3 (see definition in A293652).

%C 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 Conjecture: Sequence has infinitely many terms.

%C 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 All terms == 1 (mod 3). - _Robert Israel_, May 13 2020

%H Robert Israel, <a href="/A299704/b299704.txt">Table of n, a(n) for n = 1..2000</a>

%F 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 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 N:=2000000:

%p for X from 100 to N do

%p if isprime(X) then

%p A:=prevprime(X);

%p B:=prevprime(A);

%p C:=prevprime(B);

%p a:=X-A;

%p b:=A-B;

%p c:=B-C;

%p if a=30 and b=8 and c=6 then print(X);

%p end if

%p end if

%p end if

%p end do

%t With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Select[s, Differences@ # == {6, 8, 30} &][[All, -1]]] (* _Michael De Vlieger_, Feb 18 2018 *)

%Y Cf. A056240, A288814, A293652, A295185, A299110.

%K nonn

%O 1,1

%A _David James Sycamore_, Feb 17 2018