%I #42 Apr 02 2019 04:07:17

%S 2801,637421,2625641,78914411,195534851,7294932341,19408913261,

%T 57765899591,133311428141,212872312241,1508520377381,1960226457281,

%U 5412080545901,11543487851801,19383356741711,20748237948131,24212632812551,25413171899021,28240486488581,46922470889141

%N Brazilian primes that are also the lesser of a pair of twin primes.

%C As for Sophie Germain primes which are Brazilian (A306845), these terms are relatively rare (only 28 terms < 10^15).

%C The first 26051 terms of this sequence are of the form (11111)_b. The successive bases b are 7, 28, 40, 94, 118, 292, 373, 490, 604, 679, 1108, 1183, ... These 26051 terms end in 1: If base b ends in 1 or 6, (11111)_b ends in 5 and cannot be prime; if base b ends in another digit, then (11111)_b always ends in 1.

%C The first term which is not of this form has 31 digits; it's 1425663266336265377189900884061 = 1 + 1036 + ... + 1036^9 + 1036^10 = (11111111111)_1036 with a string of eleven 1's. In this case, the successive bases are 1036, 2089, 6961, 7894, 9775, ...

%C If (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest term for each pair (q,b) is (5,7), (11,1036), (17,1603), (23,6697), (29,2779), (41,26719), (47,98506), (53,2110).

%e 2801 is a term because 2801 + 2 = 2803 is prime, so 2801 is a lesser of twin primes, then 2801 = 1 + 7 + 7^2 + 7^3 + 7^4 = (11111)_7 and 2801 is also a Brazilian prime.

%o (PARI) lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(t+2), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ _Michel Marcus_, Mar 14 2019

%Y Intersection of A001359 and A085104.

%Y Cf. A001097, A006512, A306845, A306889.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Mar 13 2019

%E Terms computed by _Giovanni Resta_ and _Michel Marcus_, Mar 13 2019