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A306780
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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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2
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3, 1, 5, 8, 8, 8, 8, 1, 9, 3, 5, 0
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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) + ...
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EXAMPLE
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S_1 - S_2 = 0.315888819350...
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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