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A306407
Brazilian primes p such that p+2 and 2p+1 are also prime.
0
78914411, 7294932341, 119637719305001, 937391863673981, 16737518900352251, 54773061508358111, 417560366367249821, 1103799812221103741, 1515990022247085221, 2748614000294776541, 2805758307714748481, 16359900662260777211, 19024521721109192201, 126048913814465881331, 138996334987487396981
OFFSET
1,1
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.
EXAMPLE
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.
PROG
(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
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).
Sequence in context: A053431 A179578 A186598 * A210164 A119075 A136965
KEYWORD
nonn,base
AUTHOR
Bernard Schott, Apr 05 2019
STATUS
approved