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.

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.

Links

Table of n, a(n) for n=1..83.

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

Adjacent sequences: A306891 A306892 A306893 * A306895 A306896 A306897

Keyword

nonn,frac

Author

Andres Cicuttin, Mar 15 2019

Status

approved