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%I #31 Jun 09 2024 22:16:28
%S 8,2,9,3,6,5,0,1,9,7,0,2,2,2,3,3,2,0,4,9,6,2,1,9,2,4,4,3,0,8,6,1,5,4,
%T 1,6,7,3,1,5,4,8,4,7,6,2,7,5,8,3,3,9,5,5,7,3,6,4,9,0,8,2,8,9,7,7,2,8,
%U 1,9,2,1,2,3,8,7,1,4,6,6,8,3,9,2,5,8,0,0,9,6,8,5,6,9,5,1,5,5,5,8
%N Decimal expansion of 2^-1 + 2^-2 + 2^-4 + 2^-6 + 2^-10 + ..., where the exponents are 1 less than the primes.
%C Decimal expansion of Sum_{k >= 1} A010051(k)/2^(k-1).
%C The primes have a larger measure than the composites as they dominate the lower integers.
%C The binary JIS function (as defined in A113829) for this constant (that we may call the van der Waerden-Ulam constant W) is given by the first differences of A000720, A000720(n+1) - A000720(n) = A010051(n+1) = JIS[W,2]. - _Artur Jasinski_, Jun 02 2008
%D S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 54.
%F Equals 2*A051006 = 1/2 + 1/4 + 1/16 + 1/64 + 1/1024 +1/4096 + 1/65536 + ... (see A061286).
%F Equals Sum_{k>=1} pi(k)/2^k, where pi(k) = A000720(k). - _Amiram Eldar_, Aug 11 2020
%F Equals 1 - A119524. - _Antonio Graciá Llorente_, Jan 14 2024
%e 0.829365...
%t b = 0; Do[k = PrimePi[n + 1] - PrimePi[n]; b = b + k/2^n, {n, 1, 200}]; First[RealDigits[N[b, 200]]] (* _Artur Jasinski_, Jun 02 2008 *)
%o (PARI) s=0;forprime(p=2,1000,s+=1.>>p);2*s \\ _Charles R Greathouse IV_, Apr 05 2012
%Y Cf. A000720, A010051, A061286, A113829, A119524 (measure of composites).
%K nonn,cons
%O 0,1
%A _Roger L. Bagula_, May 27 2006
%E More terms from Peter Pein (petsie(AT)dordos.net), May 31 2006
%E Edited by _N. J. A. Sloane_, Nov 17 2006
%E Use of PrimePi in the first comment line corrected by _R. J. Mathar_, Oct 30 2010, _Alonso Del Arte_, Apr 05 2012
%E a(99) corrected by _Sean A. Irvine_, Jun 09 2024