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

Author

Marius Coman, Sep 19 2012

Keywords

• nonn

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

Programs

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

Mathematica

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

PARI

is(n)=n%3==1 && isprime(n\3+1) && isprime(2*n+5) \\ Charles R Greathouse IV, Dec 07 2014

Extensions

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

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