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%I #21 Jul 15 2019 21:50:00
%S 2,2,3,2,3,5,2,3,2,5,3,7,5,2,7,3,2,5,3,11,7,5,2,7,3,11,5,13,11,7,2,13,
%T 3,2,11,5,3,17,13,7,5,7,2,17,11,3,19,13,11,5,2,13,7,3,2,17,5,3,19,17,
%U 11,7,5,2,19,13,7,3,23,11,5,23,17,13,11,7,2,19,13,3,17,11,5
%N Numerator of the irreducible fractions of the form x/y, with x,y primes and x < y, sorted according to increasing k, with k = x + y. Fractions with same k are sorted in order of increasing denominator.
%C The parameter k is the Manhattan distance of the corresponding points to the origin in the "denominator, numerator" representation space.
%C This sequence (numerators) and A307701 (denominators) are respectively subsequences of A280073 and A280315, when both A280073(x) and A280315(x) are primes.
%e The first fractions are 2/3, 2/5, 3/5, 2/7, 3/7, ... with increasing k = 5,7,8,9,10,... respectively, then first terms are the corresponding numerators, 2,2,3,2,3,...
%t nmax=125;
%t (* fracs are fractions represented in the triangle with vertices (2,3),(2,prime(nmax)) and (prime(nmax),prime(nmax)) *)
%t fracs=Sort@Flatten@Table[Prime[a]/Prime[b],{b,2,nmax},{a,1,b-1}];
%t (* Sorting generated fractions according to increasing Manhattan distance first,and then by increasing denominator *)
%t fracsorted=SortBy[fracs,{Numerator@#+Denominator@#&,Denominator@#&}];
%t nmaxlimit=Floor[(1/6)*nmax^2]; (* Safe limit for a correctly sorted sequence since asymptotically half of the generated fractions can be properly sorted according to Manhattan distance *)
%t Take[Numerator@fracsorted,nmaxlimit]
%Y Cf. A307701 (denominators), A279782, A279783, A280073, A280315.
%K nonn,frac
%O 1,1
%A _Andres Cicuttin_, Mar 15 2019
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