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

1,1

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)

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A098090, A067076, A153183, A024893.

Sequence in context: A168609 A266951 A102903 * A018464 A063446 A339775

Adjacent sequences: A294061 A294062 A294063 * A294065 A294066 A294067

Keyword

nonn

Author

Dimitris Valianatos, Oct 22 2017

Status

approved