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

1,1

Links

Table of n, a(n) for n=1..38.

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A123921, A342565.

Sequence in context: A107710 A114689 A280532 * A338267 A216508 A057451

Adjacent sequences: A342561 A342562 A342563 * A342565 A342566 A342567

Keyword

nonn,easy

Author

Hugo Pfoertner, Jun 20 2021

Status

approved