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Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).
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%I #17 May 23 2019 08:34:37

%S 3,1,5,8,8,8,8,1,9,3,5,0

%N Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(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 algebraic sum compared with neighboring terms.

%F 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) + ...

%e S_1 - S_2 = 0.315888819350...

%o (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

%Y Cf. A000040, A306658 (S_1), A306700 (S_2), A306744 (S_1 + S_2).

%K cons,nonn,more

%O 0,1

%A _Marco RipĂ _ and _Aldo Roberto Pessolano_, Mar 09 2019