A275306 - OEIS

%I #37 Jan 14 2024 09:01:45

%S 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,

%T 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,

%U 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

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

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

%C 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).

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

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/primesbin.txt">Primes coded in binary to 1000 digits</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompositeNumber.html">Composite Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstant.html">Prime Constant</a>.

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

%F From _Amiram Eldar_, Aug 11 2020: (Start)

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

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

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

%F Equals Sum_{k >= 1} ((-1)^A010051(k))/2^(k+1). - _Antonio Graciá Llorente_, Jan 13 2024

%e 0.0853174901... = (0.00010101110...)_2.

%e                         | | |||

%e                         4 6 8910

%t 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 *)

%o (PARI) s=.5; forprime(p=2,bitprecision(s)+2, s-=1.>>p); s \\ _Charles R Greathouse IV_, Jul 22 2016

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

%K nonn,cons

%O 0,2

%A _Ilya Gutkovskiy_, Jul 22 2016