A161892 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).

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, 135, 36, 38, 40, 85, 179, 189, 99, 52, 217, 113, 119, 249, 261, 68, 281, 293, 76, 317, 327, 85, 355, 365, 94, 391, 407, 419, 108, 443, 455, 118, 485, 501, 517, 265, 547, 281, 144, 148

Offset: 2

Author

Daniel Tisdale, Jun 21 2009

Keywords

• nonn
• frac
• less

Comments

The sum converges rapidly to 1/2. For 100 summands, S(n) = 0.4992...; for 500, S(n) = 0.49995...

Examples

			First few fractions are 1/4, 1/3, 7/18, 9/22, 13/30, 4/9, 5/11, 23/50, 7/15, ...
		

Crossrefs

Cf. A000720 (pi), A161893 (denominators).

Cf. A161621.

Programs

Mathematica

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

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

Extensions

Offset 2 and more terms from Michel Marcus, Aug 15 2022