A342565 Numbers k such that 6*k - 1 is a prime that can be written as p*q - 2, with p and q being consecutive primes.

4265, 7842, 11265, 22815, 52265, 160065, 167662, 322003, 383542, 393722, 1016815, 1051677, 1150182, 1290842, 1372803, 1555498, 1826015, 2184065, 2808498, 3168265, 3200307, 3231062, 3333117, 3427680, 3676962, 3913915, 4042598, 4323102, 4537907, 4623542, 4798955

Offset: 1

Author

Hugo Pfoertner, Jun 20 2021

Keywords

• nonn
• easy

Examples

			a(1) = 4265, because the prime 25589 = 6*4265 - 1 can be written as 157*163 - 2, with 157 and 163 being consecutive primes.
		

Crossrefs

Cf. A123921, whose first term 13 = 3*5 - 2 cannot be written as 6*k - 1.

Cf. A048880, A342564.

Programs

PARI

a342565(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2-2);if(isprime(p),print1((p+1)/6,", "));p1=p2)};
a342565(5400)

Python

from primesieve.numpy import n_primes
from numbthy import isprime
primesarray = numpy.array(n_primes(10000, 1))
for i in range (0, 9999):
    totest = int(primesarray[i] * primesarray[i+1] - 2)
    if (isprime(totest)) and  (((totest+1)%6) == 0):
        print((totest+1)//6) # Karl-Heinz Hofmann, Jun 20 2021

Formula

a(n) = (A123921(n+1) + 1)/6, excluding A123921(1).