A245305 - OEIS

%I #21 Jan 24 2022 07:11:42

%S 1,4,7,37,142,154,202,214,307,424,469,487,499,559,577,664,742,814,847,

%T 979,982,1054,1129,1159,1162,1252,1369,1522,1612,1642,1672,1837,1987,

%U 2107,2134,2149,2209,2242,2359,2407,2419,2482,2632,2677,2767,2887,2929,2944

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

%C Sequence is infinite (Sierpiński).

%C Infinitude of the sequence would follow from Dickson's (unproved) conjecture. - _Jens Kruse Andersen_, Jul 18 2014

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

%H Reinhard Zumkeller, <a href="/A245305/b245305.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Range[0, 3000], PrimeQ[4 # + 1] && PrimeQ[4 # + 3] && PrimeQ[6 # + 5] &] (* _Vincenzo Librandi_, Jun 15 2015 *)

%o (Haskell)

%o a245305 n = a245305_list !! (n-1)

%o a245305_list = map ((`div` 4) . (subtract 1) . head) $

%o    filter (all (== 1) . map a010051') $

%o           iterate (zipWith (+) [4, 4, 6]) [1, 3, 5]

%o (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

%o (PARI) isok(k) = isprime(4*k+1) && isprime(4*k+3) && isprime(6*k+5); \\ _Michel Marcus_, Jan 24 2022

%Y Cf. A010051, A007811, A125855, A245304.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 18 2014