A306759 Decimal expansion of the sum of reciprocals of Brazilian primes, also called the Brazilian primes constant.

3, 3, 1, 7, 5, 4, 4, 6, 6

Offset: 0

Author

Bernard Schott, Mar 08 2019

Keywords

• nonn
• more
• cons

Comments

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.

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.

q S(10^q)

== ========================

1 0.1428571428571428571... (= 1/7)

2 0.2889927283868234859...

3 0.3229022355626914481...

4 0.3295236806353669357...

5 0.3312171311946179843...

6 0.3316038696349217289...

7 0.3317139158654747333...

8 0.3317434191078170412...

9 0.3317513267394988538...

10 0.3317535651668937256...

11 0.3317542057931842329...

12 0.3317543906772274268...

13 0.3317544444033188051...

14 0.3317544601136967527...

15 0.3317544647354485208...

16 0.3317544661014868080...

17 0.3317544665073451951...

18 0.3317544666282877863...

19 0.3317544666644601817...

20 0.3317544666753095766...

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.

Examples

			1/7 + 1/13 + 1/31 + 1/43 + 1/73 + 1/127 + 1/157 + ... = 0.33175...
		

References

• Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 175.

Links

• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
• Wikipedia, Goormaghtigh conjecture.

Crossrefs

Cf. A085104 (Brazilian primes), A002383 (Brazilian primes (111)_b), A225148 (Brazilian primes of the form (b^q-1)/(b-1) with q prime >= 5).

Cf. A173898 (sum of the reciprocals of the Mersenne primes), A065421 (Brun's constant).

Programs

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);
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

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

Formula

Equals Sum_{n>=1} 1/A085104(n).