A216880 - OEIS

%I #31 Mar 04 2023 16:24:32

%S 4,7,13,19,31,37,49,67,109,139,181,217,247,301,307,319,391,409,451,

%T 517,541,697,721,769,787,811,829,847,877,931,937,991,1039,1099,1117,

%U 1189,1327,1381,1399,1507,1669,1729,1777,1801,1819,1921,1957,1981,2047,2179,2251,2281

%N Numbers of the form 3p - 2 where p and 6p + 1 are prime.

%C This formula produces many primes and semiprimes.

%C Taken just the terms from the sequence above:

%C n is prime for the following values of p: 3, 5, 7, 11, 13, 23, 37, 47, 61, 103, 137, 181, 257, 263, 271, 277, 293, 313, 331, 347, 373, 443, 461, 467, 557, 593, 601, 727, 751, 761.

%C n is a semiprime of the form (6*m + 1 )*(6*n + 1) for the following values of p: 73, 83, 101, 241, 367, 653, 661.

%C n is a semiprime of the form (6*m - 1 )*(6*n - 1) for the following values of p: 107, 131, 151, 173, 397, 503, 607, 641, 683.

%C n is the square of a prime for the following values of p: 2, 17.

%C n is an absolute Fermat pseudoprime for the following value of p: 577.

%C n is a product, not squarefree, of two primes for the following values of p: 283, 311.

%C Note: any number from the sequence is a term of one of the categories above.

%C This sequence is infinite under Dickson's conjecture. - _Charles R Greathouse IV_, Sep 20 2012

%H Marius A. Burtea, <a href="/A216880/b216880.txt">Table of n, a(n) for n = 1..18020</a>

%t 3#-2&/@Select[Prime[Range[200]],PrimeQ[6#+1]&] (* _Harvey P. Dale_, Mar 04 2023 *)

%o (PARI) is(n)=n%3==1 && isprime(n\3+1) && isprime(2*n+5) \\ _Charles R Greathouse IV_, Dec 07 2014

%o (MATLAB) p=primes(10000);

%o m=1;

%o for  u=1:1000

%o     if  isprime(6*p(u)+1)==1

%o         sol(m)=3*p(u)-2;

%o         m=m+1;

%o     end

%o end

%o sol % _Marius A. Burtea_, Apr 10 2019

%o (Magma) [3*p-2:p in PrimesUpTo(1000)| IsPrime(6*p+1)]; // _Marius A. Burtea_, Apr 10 2019

%K nonn

%O 1,1

%A _Marius Coman_, Sep 19 2012

%E a(1) added, comment corrected by _Paolo P. Lava_, Dec 18 2012

%E Missing term 697 added by _Marius A. Burtea_, Apr 10 2019