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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)