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

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

Offset: 0

Author

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

Keywords

• nonn
• cons

Comments

Mingarelli shows that this constant is irrational. - Charles R Greathouse IV, Nov 05 2013

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

Examples

			0.67519843791111434190056160759135729953927678856513265156034106451688586148...
		

Links

• Angelo B. Mingarelli, Abstract factorials, arXiv:0705.4299 [math.NT], 2007-2012.

Crossrefs

Cf. A000720, A010051, A039716, A094289.

Programs

Mathematica

RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* Amiram Eldar, Nov 25 2020 *)

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

Formula

Equals Sum_{k>0} A010051(k)/k!. - R. J. Cano, Jan 25 2017

From Amiram Eldar, Nov 25 2020: (Start)

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

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