A216880 Numbers of the form 3p - 2 where p and 6p + 1 are prime.

4, 7, 13, 19, 31, 37, 49, 67, 109, 139, 181, 217, 247, 301, 307, 319, 391, 409, 451, 517, 541, 697, 721, 769, 787, 811, 829, 847, 877, 931, 937, 991, 1039, 1099, 1117, 1189, 1327, 1381, 1399, 1507, 1669, 1729, 1777, 1801, 1819, 1921, 1957, 1981, 2047, 2179, 2251, 2281

Offset

1,1

Comments

This formula produces many primes and semiprimes.

Taken just the terms from the sequence above:

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.

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.

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.

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

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

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

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

This sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Sep 20 2012

Links

Marius A. Burtea, Table of n, a(n) for n = 1..18020

Mathematica

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

Prog

(PARI) is(n)=n%3==1 && isprime(n\3+1) && isprime(2*n+5) \\ Charles R Greathouse IV, Dec 07 2014
(MATLAB) p=primes(10000);
m=1;
for  u=1:1000
    if  isprime(6*p(u)+1)==1
        sol(m)=3*p(u)-2;
        m=m+1;
    end
end
sol % Marius A. Burtea, Apr 10 2019
(Magma) [3*p-2:p in PrimesUpTo(1000)| IsPrime(6*p+1)]; // Marius A. Burtea, Apr 10 2019

Crossrefs

Sequence in context: A023496 A191868 A176003 * A144730 A058570 A134781

Adjacent sequences: A216877 A216878 A216879 * A216881 A216882 A216883

Keyword

nonn

Author

Marius Coman, Sep 19 2012

Extensions

a(1) added, comment corrected by Paolo P. Lava, Dec 18 2012

Missing term 697 added by Marius A. Burtea, Apr 10 2019

Status

approved