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Decimal expansion of Sum_{n>0} 1/prime(n)!.
6

%I #33 Nov 25 2020 09:41:05

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

%C Mingarelli shows that this constant is irrational. - _Charles R Greathouse IV_, Nov 05 2013

%C 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 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 Angelo B. Mingarelli, <a href="https://arxiv.org/abs/0705.4299">Abstract factorials</a>, arXiv:0705.4299 [math.NT], 2007-2012.

%F Equals Sum_{k>0} A010051(k)/k!. - _R. J. Cano_, Jan 25 2017

%F From _Amiram Eldar_, Nov 25 2020: (Start)

%F Equals Sum_{k>=1} 1/A039716(k).

%F Equals Sum_{k>=1} pi(k)/((k+1)*(k-1)!), where pi = A000720. (End)

%e 0.67519843791111434190056160759135729953927678856513265156034106451688586148...

%t RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* _Amiram Eldar_, Nov 25 2020 *)

%o (PARI) default(realprecision,100); sum(n=1,100,1/(prime(n)!),0.)

%o (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 Cf. A000720, A010051, A039716, A094289.

%K nonn,cons

%O 0,1

%A Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004