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A306894
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.
1
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, 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, 11, 7, 5, 2, 19, 13, 7, 3, 23, 11, 5, 23, 17, 13, 11, 7, 2, 19, 13, 3, 17, 11, 5
OFFSET
1,1
COMMENTS
The parameter k is the Manhattan distance of the corresponding points to the origin in the "denominator, numerator" representation space.
This sequence (numerators) and A307701 (denominators) are respectively subsequences of A280073 and A280315, when both A280073(x) and A280315(x) are primes.
EXAMPLE
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,...
MATHEMATICA
nmax=125;
(* fracs are fractions represented in the triangle with vertices (2, 3), (2, prime(nmax)) and (prime(nmax), prime(nmax)) *)
fracs=Sort@Flatten@Table[Prime[a]/Prime[b], {b, 2, nmax}, {a, 1, b-1}];
(* Sorting generated fractions according to increasing Manhattan distance first, and then by increasing denominator *)
fracsorted=SortBy[fracs, {Numerator@#+Denominator@#&, Denominator@#&}];
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 *)
Take[Numerator@fracsorted, nmaxlimit]
CROSSREFS
Cf. A307701 (denominators), A279782, A279783, A280073, A280315.
Sequence in context: A220370 A291048 A022467 * A169614 A037126 A080092
KEYWORD
nonn,frac
AUTHOR
Andres Cicuttin, Mar 15 2019
STATUS
approved