A293258 Decimal expansion of product of 1 - 4^-p over all primes p.

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

Offset: 0

Author

Charles R Greathouse IV, Oct 04 2017

Keywords

• nonn
• cons

Comments

Knopfmacher proves that prime(n+1) = floor(1 - log(1 - A/P)) where A is this constant and P is the product (1 - 4^-2)(1 - 4^-3)(1 - 4^-5)...(1 - 4^-prime(n)).

Examples

			0.921893835296931859194670302799807186732205478738862674976230660393864453122...
		

Links

• John Knopfmacher, Recursive formulae for prime numbers, Archiv der Mathematik (Basel) 33:2 (1979/80), pp. 144-149.

Crossrefs

Cf. A184082, A184083, A184084.

Programs

PARI

prodeuler(p=2, bitprecision(1.)/2+2, 1-4.^-p)

Formula

Equals A184082 * A184083 = A184082 / A184084. - Amiram Eldar, Nov 16 2021