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A161892 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. 1
1, 1, 7, 9, 13, 4, 5, 23, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..9.

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,...

MATHEMATICA

Table[Sum[(PrimePi[(i+1)^2]-PrimePi[i^2])/(PrimePi[(i+1)^2]*PrimePi[i^2]), {i, 2, j}], {j, 2, 50}]

CROSSREFS

Cf. A161621

Sequence in context: A121056 A174189 A112529 * A056528 A055565 A196088

Adjacent sequences:  A161889 A161890 A161891 * A161893 A161894 A161895

KEYWORD

nonn,frac

AUTHOR

Daniel Tisdale, Jun 21 2009

STATUS

approved

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Last modified August 22 00:43 EDT 2019. Contains 326169 sequences. (Running on oeis4.)