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A245305
Numbers k such that 4k+1, 4k+3, and 6k+5 are all primes.
3
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
1,2
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
MATHEMATICA
Select[Range[0, 3000], PrimeQ[4 # + 1] && PrimeQ[4 # + 3] && PrimeQ[6 # + 5] &] (* Vincenzo Librandi, Jun 15 2015 *)
PROG
(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
(PARI) isok(k) = isprime(4*k+1) && isprime(4*k+3) && isprime(6*k+5); \\ Michel Marcus, Jan 24 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 18 2014
STATUS
approved