A340469 - OEIS

%I #21 Dec 22 2023 08:29:56

%S 1,2,1,4,8,2,0,8,0,5,5,2,4,3,3,3,7,4,6,9,4,5,1,3,1,2,3,4,2,2,3,7,7,0,

%T 9,5,4,2,5,9,1,5,0,2,6,0,2,1,2,2,7,2,3,9,1,5,8,0,4,1,4,6,1,9,2,9,3,8,

%U 1,3,9,9,0,9,7,9,7,6,3,2,6,0,6,0,4,0,5,9,0,6,0,3,3,3,5,5,6,2,6,5,6,3,9,0,8

%N First constant from family of prime-representing constants h_n (h1 = 1.2148208055...) such that ceiling(h_n) = prime(n).

%C The family of constants h_n (h1 = 1.2148208055...) for generation of the complete sequence of primes with using of a recursive relation for h_n such that ceiling(h_n) = prime(n). The recursive relation h_n = ceiling(h_{n-1})*(h_{n-1}-ceiling(h_{n-1})+2) generates the complete sequence of prime numbers. Constants h_n are irrational for all n.

%H I. A. Weinstein, <a href="https://arxiv.org/abs/2101.00094">Family of prime-representing constants: use of the ceiling function</a>, arXiv:2101.00094 [math.GM], 2021.

%F h1 = Sum_{k>=1} (prime(k)-2)/Product_{i=1..k-1} prime(i).

%F Equals A249270 - A064648 - 1. - _Antonio Graciá Llorente_, Dec 22 2023

%e h1 = 1.21482080552433374694513123422377095425915026021227...

%e h2 = 2.42964161104866749389026246844754190851830052042454...

%e h3 = 4.28892483314600248167078740534262572555490156127363...

%e etc.

%t N[Sum[(Prime[k]-2)/Product[Prime[n],{n,1,k-1}],{k,1,150}],50]

%o (PARI) suminf(k=1, (prime(k)-2)/prod(i=1, k-1, prime(i))) \\ _Michel Marcus_, Jan 08 2021

%Y Cf. A000040, A249270.

%K nonn,cons

%O 1,2

%A _Ilya Weinstein_, Jan 08 2021