A245305 Numbers k such that 4k+1, 4k+3, and 6k+5 are all primes.

1, 4, 7, 37, 142, 154, 202, 214, 307, 424, 469, 487, 499, 559, 577, 664, 742, 814, 847, 979, 982, 1054, 1129, 1159, 1162, 1252, 1369, 1522, 1612, 1642, 1672, 1837, 1987, 2107, 2134, 2149, 2209, 2242, 2359, 2407, 2419, 2482, 2632, 2677, 2767, 2887, 2929, 2944

Offset: 1

Author

Reinhard Zumkeller, Jul 18 2014

Keywords

• nonn

Comments

Sequence is infinite (Sierpiński).

Infinitude of the sequence would follow from Dickson's (unproved) conjecture. - Jens Kruse Andersen, Jul 18 2014

References

• W. Sierpiński, A Selection of Problems in the Theory of Numbers. Pergamon, 1964, p. 52, #15.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A010051, A007811, A125855, A245304.

Programs

Haskell

a245305 n = a245305_list !! (n-1)
a245305_list = map ((`div` 4) . (subtract 1) . head) $
   filter (all (== 1) . map a010051') $
          iterate (zipWith (+) [4, 4, 6]) [1, 3, 5]

Magma

[n: n in [0..3*10^3] | IsPrime(4*n+1) and IsPrime(4*n+3) and IsPrime(6*n+5)]; // Vincenzo Librandi, Jun 15 2015

Mathematica

Select[Range[0, 3000], PrimeQ[4 # + 1] && PrimeQ[4 # + 3] && PrimeQ[6 # + 5] &] (* Vincenzo Librandi, Jun 15 2015 *)

PARI

isok(k) = isprime(4*k+1) && isprime(4*k+3) && isprime(6*k+5); \\ Michel Marcus, Jan 24 2022