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%I #7 Jul 09 2019 13:06:50
%S 1,3,0,4,5,6,2,5,9,8,3,5
%N Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.
%C The constant S_1* + S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.
%C On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.13045626983537 < S_1* + S_2* < 0.13045626983578.
%F S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(j + 1), prime((j + 2) - 1)} k) = 1/(3*4) + 1/(5*6) + 1/(7*8*9*10) + 1/(11*12) + ...
%e S_1* + S_2* = 0.130456269835...
%Y Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2).
%K cons,nonn,more
%O 0,2
%A _Marco Ripà_ and _Aldo Roberto Pessolano_, Apr 06 2019