A342564 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.

6, 13, 37, 73, 793, 3750, 5400, 8893, 9600, 10082, 12150, 12973, 15913, 16537, 26533, 27335, 29400, 32413, 39853, 54150, 63037, 69337, 82835, 113437, 126142, 134085, 185852, 277350, 290400, 370513, 432553, 478837, 531037, 585937, 667333, 768980, 837013, 889350

Offset: 1

Author

Hugo Pfoertner, Jun 20 2021

Keywords

• nonn
• easy

Examples

			a(1) = 6 because 6*6 + 1 = 37 can be written as 5*7 + 2.
		

Crossrefs

Cf. A048880, whose first term 17 = 3*5 + 2 cannot be written as 6*k + 1.

Cf. A123921, A342565.

Programs

Maple

alist := proc(upto) local L, q, p, n, r; L := []; q := 2;
for n from 1 to upto do
    p := q; q := nextprime(p); r := p * q + 1 ;
    if modp(r, 6) = 0 and isprime(r + 1) then
       L := [op(L), iquo(r, 6)] fi od;
L end: alist(350); # Peter Luschny, Jun 20 2021

Mathematica

(Select[6Range[10^6]+1, PrimeQ[#] && MatchQ[FactorInteger[#-2], {{p_, 1}, {q_, 1}} /; q == NextPrime[p]]&]-1)/6 (* Jean-François Alcover, Jul 07 2021 *)

PARI

a342564(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2+2);if(isprime(p),print1((p-1)/6,", "));p1=p2)};
a342564(2400)

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) = (A048880(n+1) - 1)/6, excluding A048880(1).