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A100124
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Decimal expansion of Sum_{n>0} 1/prime(n)!.
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5
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6, 7, 5, 1, 9, 8, 4, 3, 7, 9, 1, 1, 1, 1, 4, 3, 4, 1, 9, 0, 0, 5, 6, 1, 6, 0, 7, 5, 9, 1, 3, 5, 7, 2, 9, 9, 5, 3, 9, 2, 7, 6, 7, 8, 8, 5, 6, 5, 1, 3, 2, 6, 5, 1, 5, 6, 0, 3, 4, 1, 0, 6, 4, 5, 1, 6, 8, 8, 5, 8, 6, 1, 4, 8, 5, 4, 2, 4, 4, 3, 3, 4, 4, 1, 1, 4, 6, 2, 7, 2, 2, 8, 0, 2, 7, 8, 9, 5, 7, 1
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OFFSET
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0,1
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COMMENTS
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Convergence follows because A100124 < e - 2 = 0.71828... = 1/2! + 1/3! + 1/4! + 1/5! because e - 2 contains every term in A100124. The relation to e suggests a different question: is this constant not just irrational but also transcendental? - Timothy Varghese, May 07 2014
This is e times the probability that a prime is chosen from a Poisson distribution with lambda = 1. - Charles R Greathouse IV, Dec 07 2014
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LINKS
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FORMULA
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Equals Sum_{k>=1} pi(k)/((k+1)*(k-1)!), where pi = A000720. (End)
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EXAMPLE
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0.67519843791111434190056160759135729953927678856513265156034106451688586148...
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MATHEMATICA
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RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* Amiram Eldar, Nov 25 2020 *)
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PROG
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(PARI) default(realprecision, 100); sum(n=1, 100, 1/(prime(n)!), 0.)
(PARI) prec=exp(lambertw(default(realprecision)/exp(1)*log(10))+1)+9; P=s=.5; p=2; forprime(q=3, prec, P/=prod(i=p+1, q, i); s+=P; p=q); s \\ Charles R Greathouse IV, Nov 05 2013
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CROSSREFS
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KEYWORD
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AUTHOR
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Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004
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STATUS
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approved
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