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%I #84 Apr 10 2019 14:01:00
%S 3,3,1,7,5,4,4,6,6
%N Decimal expansion of the sum of reciprocals of Brazilian primes, also called the Brazilian primes constant.
%C 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 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 q S(10^q)
%C == ========================
%C 1 0.1428571428571428571... (= 1/7)
%C 2 0.2889927283868234859...
%C 3 0.3229022355626914481...
%C 4 0.3295236806353669357...
%C 5 0.3312171311946179843...
%C 6 0.3316038696349217289...
%C 7 0.3317139158654747333...
%C 8 0.3317434191078170412...
%C 9 0.3317513267394988538...
%C 10 0.3317535651668937256...
%C 11 0.3317542057931842329...
%C 12 0.3317543906772274268...
%C 13 0.3317544444033188051...
%C 14 0.3317544601136967527...
%C 15 0.3317544647354485208...
%C 16 0.3317544661014868080...
%C 17 0.3317544665073451951...
%C 18 0.3317544666282877863...
%C 19 0.3317544666644601817...
%C 20 0.3317544666753095766...
%C 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 Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 175.
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>.
%F Equals Sum_{n>=1} 1/A085104(n).
%e 1/7 + 1/13 + 1/31 + 1/43 + 1/73 + 1/127 + 1/157 + ... = 0.33175...
%o (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 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 (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 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 Cf. A173898 (sum of the reciprocals of the Mersenne primes), A065421 (Brun's constant).
%K nonn,more,cons
%O 0,1
%A _Bernard Schott_, Mar 08 2019
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