
COMMENTS

These terms point out that the conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link) was false.
Giovanni Resta has found the first counterexample of Sophie Germain prime which is Brazilian. It's the 141385th Sophie Germain prime 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)_73. The other counterexamples have been found by Michel Marcus.
These numbers are relatively rare: only 25 terms < 10^15.
The 47278 initial terms of this sequence are of the form (11111)_b. The successive bases b are 73, 94, 175, 292, 418, 979, 1228, 1429, ...
The first term which is not of this form has 32 digits, it is 14781835607449391161742645225951 = 1 + 1309 + ... + 1309^9 + 1309^10 = (11111111111)_1309 with a string of eleven 1's. In this case, the successive bases b are 1309, 1348, 2215, 2323, 2461, ...
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: (5,73), (11,1309), (17,1945), (23,20413), (29,5023), (41,9565), (47,2764) (See link Jon Grantham, Hester Graves).
Other smallest pairs (q, b) are: (53, 139492), (59, 154501), (71, 7039), (83, 9325), (89, 78028), (101, 8869), (107, 86503), (113, 89986), (131, 429226), (137, 929620), (149, 1954), (167, 175), (173, 1368025).  David A. Corneth, Mar 13 2019
