A368497 Decimal expansion of the fixed point c = S(c) of S(x) = Sum_{k>=1} (prime(k) - x) / Product_{i=1..k-1} prime(i).

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

Offset: 1

Author

Antonio Graciá Llorente, Dec 27 2023

Keywords

• nonn
• cons

Comments

S(x) = (1-x)*(1+A064648) + A249270 is linear so the fixed point is unique.

With this constant as h(1) = c, sequence h(n+1) = ceiling(h(n)) * (h(n) - ceiling(h(n)) + c) is real numbers with the property that ceiling(h(n)) = prime(n).

Examples

			1.709755124475931301268259070090809...
		

Crossrefs

Cf. A064648, A249270, A000040.

Cf. A341930 (S(3/2)), A340469 (S(2)).

Programs

PARI

solve(x=1,2,suminf(k=1,(prime(k)-x)/prod(i=1,k-1,prime(i)))-x) \\ Michal Paulovic, Dec 28 2023

Formula

Equals (A249270 + A064648 + 1)/(A064648 + 2).