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.

4327, 91621, 111697, 123001, 190027, 240997, 243517, 244291, 300277, 309667, 315937, 317827, 362137, 393517, 440131, 457087, 467587, 517861, 554167, 567097, 590071, 609571, 617917, 640771, 651727, 653311, 719101, 776551, 788071, 793591, 804157, 809491, 812431, 850177, 861391, 1007857, 1070287

Offset: 1

Author

David James Sycamore, Feb 17 2018

Keywords

• nonn

Comments

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

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).

Conjecture: Sequence has infinitely many terms.

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.

All terms == 1 (mod 3). - Robert Israel, May 13 2020

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000

Crossrefs

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

Programs

Maple

N:=2000000:
for X from 100 to N do
if isprime(X) then
A:=prevprime(X);
B:=prevprime(A);
C:=prevprime(B);
a:=X-A;
b:=A-B;
c:=B-C;
if a=30 and b=8 and c=6 then print(X);
end if
end if
end if
end do

Mathematica

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

Formula

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)).