A119523 Decimal expansion of 2^-1 + 2^-2 + 2^-4 + 2^-6 + 2^-10 + ..., where the exponents are 1 less than the primes.

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, 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, 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

Offset: 0

Author

Roger L. Bagula, May 27 2006

Keywords

• nonn
• cons

Comments

Decimal expansion of Sum_{k >= 1} A010051(k)/2^(k-1).

The primes have a larger measure than the composites as they dominate the lower integers.

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

Examples

			0.829365...
		

References

• S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 54.

Crossrefs

Cf. A000720, A010051, A061286, A113829, A119524 (measure of composites).

Programs

Mathematica

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 *)

PARI

s=0;forprime(p=2,1000,s+=1.>>p);2*s \\ Charles R Greathouse IV, Apr 05 2012

Formula

Equals 2*A051006 = 1/2 + 1/4 + 1/16 + 1/64 + 1/1024 +1/4096 + 1/65536 + ... (see A061286).

Equals Sum_{k>=1} pi(k)/2^k, where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

Equals 1 - A119524. - Antonio Graciá Llorente, Jan 14 2024

Extensions

More terms from Peter Pein (petsie(AT)dordos.net), May 31 2006

Edited by N. J. A. Sloane, Nov 17 2006

Use of PrimePi in the first comment line corrected by R. J. Mathar, Oct 30 2010, Alonso Del Arte, Apr 05 2012

a(99) corrected by Sean A. Irvine, Jun 09 2024