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

211, 631, 673, 1801, 3181, 3271, 3343, 3571, 3943, 4561, 4813, 5431, 6673, 6883, 7321, 7573, 7603, 7963, 8443, 8641, 9643, 9733, 9781, 9871, 10513, 10723, 10903, 11083, 11131, 11731, 11953, 12391, 13411, 14401, 14461, 15373, 15661, 15901, 16843, 17203, 17431, 17761, 17851, 17971, 18301, 18553, 20161, 20521, 20563, 20731

Offset: 1

Author

David James Sycamore, Feb 16 2018

Keywords

• nonn

Comments

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).

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A056240, A288814, A293652, A295185.

Programs

Maple

N:=21000:
for X from 2 to N do
if isprime(X) then
A:=prevprime(X);
B:=prevprime(A);
a:=X-A;
b:=A-B;
if a=12 and b=2 then print(X);
end if
end if
end if
end do
# alternative:
P:= select(isprime, {seq(i,i=3..10^6,2)}):
Q:= P intersect map(t -> t-12, P) intersect map(t -> t+2, P):
Q:= remove(t -> ormap(isprime, [seq(t+i,i=2..10,2)]), Q):
map(t -> t+12, Q); # Robert Israel, Feb 16 2018

Mathematica

Select[Partition[Prime[Range[2500]],3,1],Differences[#]=={2,12}&][[All,3]] (* Harvey P. Dale, Feb 29 2020 *)

PARI

isok(p) =  isprime(p) && (pp=precprime(p-1)) && (p-pp == 12) && (ppp=precprime(pp-1)) && (pp-ppp == 2); \\ Michel Marcus, Feb 16 2018

Formula

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)).