A294064 Numbers k such that 2*k - 3, 2*k + 3, 3*k - 2, 3*k + 2 are primes.

5, 7, 13, 35, 43, 55, 77, 127, 133, 155, 167, 253, 287, 295, 365, 475, 497, 533, 595, 713, 1007, 1177, 1483, 1805, 2323, 2575, 2723, 2927, 3107, 3415, 3487, 3823, 4145, 4213, 4367, 4565, 4717, 4927, 4963, 5125, 5215, 5363, 5417, 5587, 5627, 5795, 6133, 6587, 6797

Offset: 1

Author

Dimitris Valianatos, Oct 22 2017

Keywords

• nonn

Comments

The common numbers of A098090, A067076, A153183, A024893.

Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges.

Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*k - 3) + (2*k + 3) + (3*k - 2) + (3*k + 2) = 10*k.

From Robert G. Wilson v, Nov 19 2017: (Start)

Number of terms less than 10^m: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ;

Number of prime terms less than 10^m: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., .

All terms are == {5, 7, 13, 17, 23, 25} (mod 30).

(End)

Examples

			5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers.
7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A098090, A067076, A153183, A024893.

Programs

Mathematica

Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* Michael De Vlieger, Oct 22 2017 *)

PARI

{
for(n=1,10000,
    if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2),
       print1(n", ")
      )
   )
}