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A161892
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Numerators of the partial sums of S(n) = Sum[(Pi((n+1)^2) - Pi(n^2))/(Pi((n+1)^2)*Pi(n^2)), n from 2 to oo. Pi(n) = number of primes <= n.
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1
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OFFSET
| 1,3
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COMMENTS
| The sum converges rapidly to 1/2. For 100 summands, S(n)= 0.4992...; for 500, S(n) = 0.49995...S(n) generalizes to: S_k(n)= Sum[(Pi((n+k)^2)-Pi((n+k-1)^2))/(Pi((n+k)^2)*Pi((n+k-1)^2))),n=2 to oo, k =1,2,3,...For k =1, the analogous series for C(n),composites <= n, appears to converge to 1.
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EXAMPLE
| First few terms of sequence are: 1/4,1/3,7/18,9/22,13/30,4/9,5/11,23/50,7/15,...
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MATHEMATICA
| Table[Sum[(PrimePi[(i+1)^2]-PrimePi[i^2])/(PrimePi[(i+1)^2]*PrimePi[i^2]), {i, 2, j}], {j, 2, 50}]
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CROSSREFS
| Cf. A161621
Sequence in context: A121056 A174189 A112529 * A056528 A055565 A196088
Adjacent sequences: A161889 A161890 A161891 * A161893 A161894 A161895
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KEYWORD
| nonn,frac
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AUTHOR
| Daniel Tisdale (daniel6874(AT)gmail.com), Jun 21 2009
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