A306849 Brazilian primes that are also the lesser of a pair of twin primes.

2801, 637421, 2625641, 78914411, 195534851, 7294932341, 19408913261, 57765899591, 133311428141, 212872312241, 1508520377381, 1960226457281, 5412080545901, 11543487851801, 19383356741711, 20748237948131, 24212632812551, 25413171899021, 28240486488581, 46922470889141

Offset: 1

Author

Bernard Schott, Mar 13 2019

Keywords

• nonn
• base

Comments

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

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.

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, ...

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).

Examples

			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.
		

Crossrefs

Intersection of A001359 and A085104.

Cf. A001097, A006512, A306845, A306889.

Programs

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

Extensions

Terms computed by Giovanni Resta and Michel Marcus, Mar 13 2019