A306407 Brazilian primes p such that p+2 and 2p+1 are also prime.

78914411, 7294932341, 119637719305001, 937391863673981, 16737518900352251, 54773061508358111, 417560366367249821, 1103799812221103741, 1515990022247085221, 2748614000294776541, 2805758307714748481, 16359900662260777211, 19024521721109192201, 126048913814465881331, 138996334987487396981

Offset: 1

Author

Bernard Schott, Apr 05 2019

Keywords

• nonn
• base

Comments

The initial terms of this sequence are of the form (11111)_b. The successive bases b are 94, 292, 3307, 5533, 11374, ...

The first term which is not of this form has 43 digits: it is 1137259672818014782224246589454763146442851 = 1 + 16054 + ... + 16054^9 + 16054^10 = (11111111111)_16054 with a string of eleven 1's.

Sophie Germain primes and lesser twins which are Brazilian both have the same property: if p = (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest terms for the first pairs (q,b) are (5,94), (11,16054), (17,3247).

Intersection of A306845 and A306849.

Intersection of A045536 and A085104.

Examples

			The prime 78914411 is a term, because 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 is a Brazilian prime, 78914411 + 2 = 78914413 is prime and 2 * 78914411 + 1 = 157828823 is prime. The prime 78914411 is Brazilian, the lesser of a pair of twin primes and also a Sophie Germain prime.
		

Crossrefs

Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).

Cf. A045536 (intersection of A001359 and A005384).

Cf. A085104 (Brazilian primes).

Cf. A306845 (Sophie Germain Brazilian primes), A306849 (lesser of twin primes which is Brazilian).

Programs

PARI

brazilp(N)=forprime(K=5, #binary(N+1)-1, for(n=4, sqrtnint(N-1, K-1), if((K%6==5)&&(n%3==1),if(isprime((n^K-1)/(n-1))&&isprime((n^K-1)/(n-1)+2)&&isprime(2*(n^K-1)/(n-1)+1), print1((n^K-1)/(n-1), ", "))))) \\ Davis Smith, Apr 06 2019