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%I #12 Sep 08 2022 08:45:45
%S 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,
%T 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,
%U 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
%N Denominators of the ratios (in lowest terms) of numbers of primes in one square interval to that of the interval and its successor.
%C The numerators are derived from sequence A014085.
%C The expression is: R(n) = (PrimePi((n+1)^2) - PrimePi(n^2))/(PrimePi((n+2)^2) - PrimePi(n^2)).
%C The first few ratios are 1/2, 2/5, 3/5, 1/3, 4/7, ...
%C Conjecture: lim_{n->infinity} R(n) = 1/2. See also more extensive comment entered with sequence of numerators. This conjecture implies Legendre's conjecture.
%e 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
%o (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
%Y Cf. A014085.
%Y Cf. A161621 (numerators). - _Klaus Brockhaus_, Jun 15 2009
%K nonn,frac
%O 1,1
%A _Daniel Tisdale_, Jun 14 2009
%E a(1) inserted and extended beyond a(11) by _Klaus Brockhaus_, Jun 15 2009
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