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

Author

Daniel Tisdale, Jun 14 2009

Keywords

• nonn
• frac

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.

Examples

			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
		

Crossrefs

Cf. A014085.

Cf. A161621 (numerators). - Klaus Brockhaus, Jun 15 2009

Programs

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

Extensions

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