OFFSET
0,1
COMMENTS
The constant S_1 - S_2 is related to the prime gaps, since twin primes produce the largest terms of the algebraic sum compared with neighboring terms.
FORMULA
S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!) = Sum_{j>=2} (-1)^j/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 - 1/(4*5) + 1/(6*7) - 1/(8*9*10*11) + ...
EXAMPLE
S_1 - S_2 = 0.315888819350...
PROG
(PARI) sumalt(j=1, (-1)^(j+1)*(prime(j)!/prime(j + 1)!)) \\ Michel Marcus, Apr 02 2019 \\ Needs default(realprecision, 10^4) Jinyuan Wang, May 19 2019
CROSSREFS
KEYWORD
AUTHOR
Marco Ripà and Aldo Roberto Pessolano, Mar 09 2019
STATUS
approved