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A306780
Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).
2
3, 1, 5, 8, 8, 8, 8, 1, 9, 3, 5, 0
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
Cf. A000040, A306658 (S_1), A306700 (S_2), A306744 (S_1 + S_2).
Sequence in context: A176105 A094353 A298662 * A209996 A340804 A129801
KEYWORD
cons,nonn,more
AUTHOR
STATUS
approved