OFFSET
0,1
COMMENTS
Derived from an open problem published in 2012, the constant S_1 is connected to the gap between the j-th and (j+1)-th primes.
Together with the constant S_2 (see A306700), S_1 involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.
On Mar 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get:
0.367555442019264 < S_1 < 0.367555442019265 + Sum_{j>=2100000000} 1/((2*j*log(2*j) + log(log(2*j)) - 1) * (2*j*log(2*j) + log(log(2*j)) - 2)) < 0.367555442019265 + 1/1.94576*10^(-13) < 0.367555442019460.
REFERENCES
M. Ripà, Congetture su interrogativi inediti: tra speculazioni, voli pindarici e riflessioni spicciole, Jun 2012. ISBN 9788863699463
LINKS
Wikipedia, Rosser's theorem
FORMULA
Sum_{j>=1} prime(2*j - 1)!/prime(2*j)! = 1/3 + Sum_{j>=2} 1/(Product{k=prime(2*j - 1) + 1, prime(2*j)} k) = 1/3 + 1/(7*6) + 1/(13*12) + ...
EXAMPLE
S_1 = 0.36755544201926...
MATHEMATICA
a = 0; Do[f = Prime[Range[n - 999999, n]]; Do[a += N[1/Product[k, {k, f[[i]] + 1, f[[i + 1]]}], 100], {i, 1, 1000000, 2}]; Print[n, ": ", N[a, 100]], {n, 1000000, 100000000, 1000000}]; a
PROG
(PARI) suminf(j=1, prime(2*j - 1)!/prime(2*j)!) \\ Michel Marcus, Apr 02 2019
CROSSREFS
KEYWORD
AUTHOR
Marco Ripà, Mar 03 2019
STATUS
approved