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%I #20 May 23 2019 08:31:18
%S 4,1,9,2,2,2,0,6,4,9,0,3
%N Decimal expansion of the constant S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 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.
%F S_1 + S_2 = Sum_{j>=1} prime(j)!/prime(j + 1)! = Sum_{j>=2} 1/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 + 1/(4*5) + 1/(6*7) + 1/(8*9*10*11) + ...
%e S_1 + S_2 = 0.419222064903...
%t s = 0; p = 2; q = 3; While[p < 10^10, s = N[s + 1/Times @@ Range[p +1, q], 32]; p = q; q = NextPrime@ q]; Take[ RealDigits[s][[1]], 20] (* _Robert G. Wilson v_, Mar 23 2019 *)
%o (PARI) suminf(j=1, prime(j)!/prime(j + 1)!) \\ _Michel Marcus_, Apr 02 2019
%Y Cf. A000040, A306658 (S_1), A306700 (S_2), A306780.
%K cons,nonn,more
%O 0,1
%A _Marco Ripà_, Mar 07 2019
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