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%I #18 Aug 15 2022 10:31:12
%S 1,1,7,9,13,4,5,23,7,8,37,21,23,13,59,16,35,19,83,45,97,103,28,30,127,
%T 135,36,38,40,85,179,189,99,52,217,113,119,249,261,68,281,293,76,317,
%U 327,85,355,365,94,391,407,419,108,443,455,118,485,501,517,265,547,281,144,148
%N Numerators of S(n) = Sum_{j=2..n} (pi((j+1)^2) - pi(j^2))/(pi((j+1)^2)*pi(j^2)) where pi(k) = A000720(k).
%C The sum converges rapidly to 1/2. For 100 summands, S(n) = 0.4992...; for 500, S(n) = 0.49995...
%e First few fractions are 1/4, 1/3, 7/18, 9/22, 13/30, 4/9, 5/11, 23/50, 7/15, ...
%t Table[Sum[(PrimePi[(i+1)^2]-PrimePi[i^2])/(PrimePi[(i+1)^2]*PrimePi[i^2]),{i,2,j}],{j,2,50}]
%o (PARI) a(n) = numerator(sum(k=2, n, (primepi((k+1)^2) - primepi(k^2))/(primepi((k+1)^2)*primepi(k^2)))); \\ _Michel Marcus_, Aug 15 2022
%Y Cf. A000720 (pi), A161893 (denominators).
%Y Cf. A161621.
%K nonn,frac,less
%O 2,3
%A _Daniel Tisdale_, Jun 21 2009
%E Offset 2 and more terms from _Michel Marcus_, Aug 15 2022
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