OFFSET
0,2
COMMENTS
The constant S_2 is connected to the gap between the j-th and (j+1)-th primes.
Together with the constant S_1 (see A306658), S_2 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.
On Mar 07 2019, the first 2445000000 prime numbers were used and from Rosser's theorem we obtain:
0.05166662288423 < S_2 < 0.05166662288424 + Sum_{j>=1222500000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.05166662288424 + 3.22757*10^(-13) < 0.05166662288457.
LINKS
Wikipedia, Rosser's theorem
FORMULA
Sum_{j>=1} prime(2*j)!/prime(2*j + 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j) + 1, prime(2*j + 1)} k) = 1/(5*4) + 1/(11*10*9*8) + 1/(17*16*15*14) + ...
EXAMPLE
S_2 = 0.0516666228842...
MATHEMATICA
b = 0; Do[f = Prime[Range[n - 999999, n]]; Do[b += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[b, 100]], {n, 1000001, 100000001, 1000000}]; b
PROG
(PARI) suminf(j=1, prime(2*j)!/prime(2*j + 1)!) \\ Michel Marcus, Apr 02 2019
CROSSREFS
KEYWORD
AUTHOR
Marco Ripà, Mar 05 2019
STATUS
approved