A307384 - OEIS

%I #7 Jul 09 2019 13:06:58

%S 0,8,5,1,6,1,9,1,0,9,8,5

%N Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2*j) - 1)!/prime((2*j + 1) - 1)!.

%C Together with the constant S_2* and S_1* + S_2* (see A307383), S_1* involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.

%C On Apr 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get: 0.08516191098523 < S_1* < 0.08516191098543.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rosser%27s_theorem">Rosser's theorem</a>

%F S_1* = Sum_{j>=1} prime(2*j - 1)!/prime((2*j + 1) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j), prime(2*j + 1)} k) = 1/(3*4) + 1/(7*8*9*10) + 1/(13*14*15*16) + 1/(19*20*21*22) +...

%e S_1* = 0.085161910985...

%Y Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*).

%K cons,nonn,more

%O 0,2

%A _Marco Ripà_ and _Aldo Roberto Pessolano_, Apr 06 2019