A340469 First constant from family of prime-representing constants h_n (h1 = 1.2148208055...) such that ceiling(h_n) = prime(n).

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, 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, 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

Offset: 1

Author

Ilya Weinstein, Jan 08 2021

Keywords

• nonn
• cons

Comments

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.

Examples

			h1 = 1.21482080552433374694513123422377095425915026021227...
h2 = 2.42964161104866749389026246844754190851830052042454...
h3 = 4.28892483314600248167078740534262572555490156127363...
etc.
		

Links

• I. A. Weinstein, Family of prime-representing constants: use of the ceiling function, arXiv:2101.00094 [math.GM], 2021.

Crossrefs

Cf. A000040, A249270.

Programs

Mathematica

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

PARI

suminf(k=1, (prime(k)-2)/prod(i=1, k-1, prime(i))) \\ Michel Marcus, Jan 08 2021

Formula

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

Equals A249270 - A064648 - 1. - Antonio Graciá Llorente, Dec 22 2023