This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306845 #57 Mar 15 2019 02:42:13 %S A306845 28792661,78914411,943280801,7294932341,30601685951,919548423641, %T A306845 2275869057821,4172851565741,4801096143881,27947620155401, %U A306845 29586967653101,43573806645461,119637719305001,124484682222941,148908227169101,172723673300501 %N A306845 Sophie Germain primes which are Brazilian. %C A306845 These terms point out that the conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link) was false. %C A306845 _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_. %C A306845 These numbers are relatively rare: only 25 terms < 10^15. %C A306845 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, ... %C A306845 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, ... %C A306845 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). %C A306845 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 %H A306845 Jon Grantham, Hester Graves, <a href="https://arxiv.org/abs/1903.04577">Brazilian Primes Which Are Also Sophie Germain Primes</a>, arXiv:1903.04577 [math.NT], 2019. %H A306845 Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature. %e A306845 78914411 is a term because 2 * 78914411 + 1 = 157828823 is prime, so 78914411 is Sophie Germain prime, then, 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 = (11111)_94 and 78914411 is also a Brazilian prime. %o A306845 (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(2*t+1), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ _Michel Marcus_, Mar 13 2019 %Y A306845 Intersection of A005384 and A085104. %Y A306845 Cf. A007528, A306849. %K A306845 nonn %O A306845 1,1 %A A306845 _Bernard Schott_, Mar 13 2019