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Primes p such that p == 5 (mod 6) and p+1 is squarefree.
1

%I #9 Dec 30 2020 03:35:13

%S 5,29,41,101,113,137,173,257,281,317,353,389,401,461,509,569,617,641,

%T 653,677,761,797,821,857,929,941,977,1109,1181,1193,1217,1229,1289,

%U 1301,1361,1373,1409,1433,1481,1553,1613,1697,1721,1877,1901,1913,1973,2081,2129

%N Primes p such that p == 5 (mod 6) and p+1 is squarefree.

%C Clary and Fabrykowski (2004) proved that this sequence is infinite, and that its relative density in the sequence of primes of the form 6*k+5 (A007528) is 4*A/5 = 0.29916465..., where A is Artin's constant (A005596).

%H Amiram Eldar, <a href="/A340154/b340154.txt">Table of n, a(n) for n = 1..10000</a>

%H Stuart Clary and Jacek Fabrykowski, <a href="https://dml.cz/handle/10338.dmlcz/127940">Arithmetic progressions, prime numbers, and squarefree integers</a>, Czechoslovak Mathematical Journal, Vol. 54, No. 4 (2004), pp. 915-927.

%e 5 is a term since it is prime, 5 == 5 (mod 6), and 5+1 = 6 = 2*3 is squarefree.

%t Select[Range[2000], Mod[#, 6] == 5 && PrimeQ[#] && SquareFreeQ[# + 1] &]

%Y Intersection of A007528 and A049097.

%Y Cf. A005117, A005596.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Dec 29 2020