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

0,1

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

References

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

Links

Table of n, a(n) for n=0..99.

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

Example

0.829365...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A019865 A198993 A307565 * A181164 A154212 A155035

Adjacent sequences: A119520 A119521 A119522 * A119524 A119525 A119526

Keyword

nonn,cons

Author

Roger L. Bagula, May 27 2006

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

Status

approved