A306407 - OEIS

%I #56 Apr 10 2019 13:58:06

%S 78914411,7294932341,119637719305001,937391863673981,

%T 16737518900352251,54773061508358111,417560366367249821,

%U 1103799812221103741,1515990022247085221,2748614000294776541,2805758307714748481,16359900662260777211,19024521721109192201,126048913814465881331,138996334987487396981

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

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

%C 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.

%C 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).

%C Intersection of A306845 and A306849.

%C Intersection of A045536 and A085104.

%e 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.

%o (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

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

%Y Cf. A045536 (intersection of A001359 and A005384).

%Y Cf. A085104 (Brazilian primes).

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

%K nonn,base

%O 1,1

%A _Bernard Schott_, Apr 05 2019