A161622 Denominators of the ratios (in lowest terms) of numbers of primes in one square interval to that of the interval and its successor.

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

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)) = (6-4)/(9-4) = 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