

A161622


Denominators of the ratios (in lowest terms) of numbers of primes in one square interval to that of the interval and its successor.


4



2, 2, 5, 5, 3, 7, 7, 7, 8, 9, 9, 2, 9, 5, 13, 12, 11, 2, 13, 2, 2, 13, 5, 17, 15, 15, 17, 17, 2, 9, 19, 19, 19, 19, 19, 2, 7, 23, 23, 23, 20, 7, 23, 24, 23, 23, 28, 5, 21, 26, 31, 7, 25, 24, 23, 29, 30, 29, 2, 29, 30, 32, 29, 15, 31, 2, 32, 30, 34, 12, 2, 32, 2, 35, 20, 18, 16, 41, 36, 33
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OFFSET

1,1


COMMENTS

The numerators are derived from sequence A014085.
The expression is: R(n) = (PrimePi((n+1)^2)  PrimePi(n^2))/(PrimePi((n+2)^2)  PrimePi(n^2)).
The first few ratios are 1/2, 2/5, 3/5, 1/3, 4/7, ...
Conjecture: lim_{n>infinity} R(n) = 1/2. See also more extensive comment entered with sequence of numerators. This conjecture implies Legendre's conjecture.


LINKS

Table of n, a(n) for n=1..80.


EXAMPLE

R(3) = (PrimePi(4^2)PrimePi(3^2)) / (PrimePi(5^2)PrimePi(3^2)) = (PrimePi(16)PrimePi(9)) / (PrimePi(25)PrimePi(9)) = (64)/(94) = 2/5. Hence a(3) = 5.  Klaus Brockhaus, Jun 15 2009


PROG

(MAGMA) [ Denominator((#PrimesUpTo((n+1)^2)  a) / (#PrimesUpTo((n+2)^2)  a)) where a is #PrimesUpTo(n^2): n in [1..80] ]; // Klaus Brockhaus, Jun 15 2009


CROSSREFS

Cf. A014085.
Cf. A161621 (numerators).  Klaus Brockhaus, Jun 15 2009
Sequence in context: A253600 A045537 A243941 * A116559 A210802 A257943
Adjacent sequences: A161619 A161620 A161621 * A161623 A161624 A161625


KEYWORD

nonn,frac


AUTHOR

Daniel Tisdale, Jun 14 2009


EXTENSIONS

a(1) inserted and extended beyond a(11) by Klaus Brockhaus, Jun 15 2009


STATUS

approved



