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A119523
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Decimal expansion of 2^-1 + 2^-2 + 2^-4 + 2^-6 + 2^-10 + ..., where the exponents are 1 less than the primes.
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5
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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, 9
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 54
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LINKS
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Table of n, a(n) for n=0..99.
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FORMULA
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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
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EXAMPLE
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0.829365...
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MATHEMATICA
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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 *)
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PROG
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(PARI) s=0; forprime(p=2, 1000, s+=1.>>p); 2*s \\ Charles R Greathouse IV, Apr 05 2012
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CROSSREFS
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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
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KEYWORD
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nonn,cons
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AUTHOR
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Roger L. Bagula, May 27 2006
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EXTENSIONS
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More terms from Peter Pein (petsie(AT)dordos.net), May 31 2006
Edited by N. J. A. Sloane, Nov 17 2006
Corrected use of PrimePi in the first comment line - R. J. Mathar, Oct 30 2010, Alonso Del Arte, Apr 05 2012
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STATUS
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approved
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