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Take the list of positive rationals {R(n): n>=1} in the order defined by Calkin and Wilf (Recounting the Rationals, 1999); a(n) = denominator of R(prime(n)).
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%I #10 Aug 19 2015 16:57:53

%S 2,1,2,1,2,3,4,3,2,4,1,7,8,5,2,8,4,5,5,4,11,3,8,12,9,12,5,8,11,10,1,6,

%T 14,9,18,7,13,11,8,18,12,19,2,11,16,7,13,3,10,17,18,4,13,6,8,6,16,5,

%U 23,22,13,26,17,10,23,16,19,29,18,23,22,12,7,25,11,2,20,23,26,29,18,31,8,27,11,14,16,27,24,7,18,4,9,14,11,6,8,20,13,21,19,32,22,30,17,23,26,40,18,43,7,41,44,27,13,20,17,14,36,30,49,37,50,34,31,28,39,12,19,33,23,16,9,31,24,15,24,25,30,50,31,46,17,22,27,18,55,50,29,8,41,36,25,14,23,10,17,32,47,40,26,34,13,22,32,14,5,27

%N Take the list of positive rationals {R(n): n>=1} in the order defined by Calkin and Wilf (Recounting the Rationals, 1999); a(n) = denominator of R(prime(n)).

%C The list of rationals {R(n)} is essentially given by A002487(n)/A002487(n+1).

%H N. Calkin and H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/website/recounting.pdf">Recounting the rationals</a>, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

%Y Companion sequence to A261179.

%K nonn

%O 1,1

%A _James Kirk Winkler_, Aug 13 2015