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Numerators in the enumeration of the rationals by Czyz and Self.
2

%I #29 Aug 19 2020 02:57:34

%S 0,1,1,2,3,1,2,3,5,4,5,1,2,3,3,4,7,7,8,5,7,7,8,1,2,3,3,4,5,4,5,5,9,10,

%T 11,9,12,11,13,6,9,10,11,9,12,11,13,1,2,3,3,4,5,4,5,5,7,7,8,5,7,7,8,6,

%U 11,13,14,13,17,15,18,11,16,17,19,14,19,18,21

%N Numerators in the enumeration of the rationals by Czyz and Self.

%C Denominators are A071766(n) for n >= 1.

%C Differs from A229742 by 1 at the integer rational positions n = 2^k because Czyz and Self only increment the last continued fraction term when there are two or more terms. So a(n) = A229742(n) - A209229(n) for n >= 1.

%H Rémy Sigrist, <a href="/A238837/b238837.txt">Table of n, a(n) for n = 1..10000</a>

%H Jerzy Czyz and William Self, <a href="http://www.jstor.org/stable/3595818">The Rationals Are Countable: Euclid's Proof</a>, The College Mathematics Journal, volume 34, number 5, November 2003, pages 367-369.

%o (PARI) a(n) = my (w=[]); while (n, my (v=valuation(n,2)); w=concat(w, 1+v); n \= 2^(v+1)); w[#w]--; my (r=w[1] + (#w>1)); for (k=2, #w, r=w[k]+1/r); numerator(r) \\ _Rémy Sigrist_, Aug 25 2018

%Y Cf. A071766, A209229, A229742.

%K nonn,frac,look

%O 1,4

%A _N. J. A. Sloane_, Mar 14 2014, following a suggestion from _Kevin Ryde_

%E More terms from _Rémy Sigrist_, Aug 25 2018