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 A306759 #84 Apr 10 2019 14:01:00
%S A306759 3,3,1,7,5,4,4,6,6
%N A306759 Decimal expansion of the sum of reciprocals of Brazilian primes, also called the Brazilian primes constant.
%C A306759 The name "constant of Brazilian primes" is used in the article "Les nombres brésiliens" in link, théorème 4, page 36. Brazilian primes are in A085104.
%C A306759 Let S(k) be the sum of reciprocals of Brazilian primes < k. These values below come from different calculations by Jon, Michel, Daniel and Davis.
%C A306759 q S(10^q)
%C A306759 == ========================
%C A306759 1 0.1428571428571428571... (= 1/7)
%C A306759 2 0.2889927283868234859...
%C A306759 3 0.3229022355626914481...
%C A306759 4 0.3295236806353669357...
%C A306759 5 0.3312171311946179843...
%C A306759 6 0.3316038696349217289...
%C A306759 7 0.3317139158654747333...
%C A306759 8 0.3317434191078170412...
%C A306759 9 0.3317513267394988538...
%C A306759 10 0.3317535651668937256...
%C A306759 11 0.3317542057931842329...
%C A306759 12 0.3317543906772274268...
%C A306759 13 0.3317544444033188051...
%C A306759 14 0.3317544601136967527...
%C A306759 15 0.3317544647354485208...
%C A306759 16 0.3317544661014868080...
%C A306759 17 0.3317544665073451951...
%C A306759 18 0.3317544666282877863...
%C A306759 19 0.3317544666644601817...
%C A306759 20 0.3317544666753095766...
%C A306759 According to the Goormaghtigh conjecture, there are only two Brazilian primes which are twice Brazilian: 31 = (111)_5 = (11111)_2 and 8191 = (111)_90 = (1111111111111)_2. The reciprocals of these two numbers are counted only once in the sum.
%D A306759 Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 175.
%H A306759 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.
%H A306759 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>.
%F A306759 Equals Sum_{n>=1} 1/A085104(n).
%e A306759 1/7 + 1/13 + 1/31 + 1/43 + 1/73 + 1/127 + 1/157 + ... = 0.33175...
%o A306759 (PARI) brazil(N, L=List())=forprime(K=3, #binary(N+1)-1, for(n=2, sqrtnint(N-1, K-1), if(isprime((n^K-1)/(n-1)),listput(L, (n^K-1)/(n-1))))); Set(L);
%o A306759 brazilcons(lim,nbd) = r=brazil(10^lim); x=sum(M=1, #r, 1./r[M]);for(n=1, nbd, print1(floor(x*10^n)%10, ", "));\\ _Davis Smith_, Mar 10 2019
%o A306759 (PARI) cons(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t), listput(v, t)))); v = vecsort(Vec(v), , 8); sum(k=1, #v, 1./v[k]); \\ _Michel Marcus_, Mar 11 2019
%Y A306759 Cf. A085104 (Brazilian primes), A002383 (Brazilian primes (111)_b), A225148 (Brazilian primes of the form (b^q-1)/(b-1) with q prime >= 5).
%Y A306759 Cf. A173898 (sum of the reciprocals of the Mersenne primes), A065421 (Brun's constant).
%K A306759 nonn,more,cons
%O A306759 0,1
%A A306759 _Bernard Schott_, Mar 08 2019