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A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.

%I A100124 #33 Nov 25 2020 09:41:05
%S A100124 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,
%T A100124 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,
%U A100124 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
%N A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.
%C A100124 Mingarelli shows that this constant is irrational. - _Charles R Greathouse IV_, Nov 05 2013
%C A100124 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
%C A100124 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
%H A100124 Angelo B. Mingarelli, <a href="https://arxiv.org/abs/0705.4299">Abstract factorials</a>, arXiv:0705.4299 [math.NT], 2007-2012.
%F A100124 Equals Sum_{k>0} A010051(k)/k!. - _R. J. Cano_, Jan 25 2017
%F A100124 From _Amiram Eldar_, Nov 25 2020: (Start)
%F A100124 Equals Sum_{k>=1} 1/A039716(k).
%F A100124 Equals Sum_{k>=1} pi(k)/((k+1)*(k-1)!), where pi = A000720. (End)
%e A100124 0.67519843791111434190056160759135729953927678856513265156034106451688586148...
%t A100124 RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* _Amiram Eldar_, Nov 25 2020 *)
%o A100124 (PARI) default(realprecision,100); sum(n=1,100,1/(prime(n)!),0.)
%o A100124 (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
%Y A100124 Cf. A000720, A010051, A039716, A094289.
%K A100124 nonn,cons
%O A100124 0,1
%A A100124 Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004