A275306 Decimal expansion of 1/2 - Sum_{k>=1} 1/2^prime(k).

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

Offset: 0

Author

Ilya Gutkovskiy, Jul 22 2016

Keywords

• nonn
• cons

Comments

Composite constant: decimal value of A066247 interpreted as a binary number.

The characteristic function of composite numbers (A066247) has values 0, 0, 0, 1, 0, 1, 0, 1, 1, ... for n = 1, 2, 3, ... The constant obtained by concatenating these digits and interpreting them as a binary fraction is therefore C = 0.0001010111010... (base 2) = 0.0853174901...(base 10).

Continued fraction [0; 11, 1, 2, 1, 1, 2, 1, 1, 131, 2, 1, 1, 2, 6, 4, 2, 21, ...].

Examples

			0.0853174901... = (0.00010101110...)_2.
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Links

• Simon Plouffe, Primes coded in binary to 1000 digits.
• Eric Weisstein's World of Mathematics, Composite Number.
• Eric Weisstein's World of Mathematics, Prime Constant.

Crossrefs

Cf. A002808, A051006, A005171, A062298, A066247, A073718.

Programs

Mathematica

nn = 121; Take[#, nn] &@ PadLeft[First@ #, Abs@ Last@ # + Length@ First@ #] &@ RealDigits@ N[1/2 - Sum[ 1/2^Prime[k], {k, 10^4}], nn + 2] (* Michael De Vlieger, Jul 22 2016 *)

PARI

s=.5; forprime(p=2,bitprecision(s)+2, s-=1.>>p); s \\ Charles R Greathouse IV, Jul 22 2016

Formula

Equals Sum_{k>=1} 1/2^A002808(k).

From Amiram Eldar, Aug 11 2020: (Start)

Equals Sum_{k>=1} 1/A073718(k).

Equals Sum_{k>=1} A066247(k)/2^k.

Equals -(1/2) + Sum_{k>=1} A062298(k)/2^(k+1). (End)

Equals Sum_{k >= 1} ((-1)^A010051(k))/2^(k+1). - Antonio Graciá Llorente, Jan 13 2024