A279369 A mapping of rationals a/b (lowest form) to prime rationals p/q such that a/b = (p+1)/(q+1), where n (the sequence index) selects the rationals a/b from the triangle array A226314(n)/A054531(n) and a(n) selects the prime rationals p/q from the same array.

1, 12, 18, 58, 13, 74, 57, 19, 5, 72, 174, 178, 182, 429, 217, 138, 8, 225, 247, 272, 162, 825, 81, 83, 85, 849, 89, 999, 255, 1047, 23, 110, 484, 103, 1122, 288, 1383, 139, 114, 143, 1407, 32, 149, 1425, 1518, 408, 711, 176, 1677, 165, 727, 184, 1701, 188, 450, 906, 910, 914

Offset

1,2

Comments

Rationals a/b (lowest form) can be mapped 1-to-1 to a positive integer n where a/b is the n-th term of the triangular array A226314(n)/A054531(n). Consider two function of x, f_1 = ax-1 and f_2 = bx-1. Then by Schinzel's Hypothesis H there are infinite values of x such that f_1 and f_2 are simultaneously prime allowing a/b to be expressed using two primes p and q as a/b=(p+1)/(q+1).

By choosing the least x for generating p=f_1 and q=f_2 (see A278635) it is possible to find a unique prime rational p/q that maps to rational a/b. If n is the sequence index that selects the rational a/b from the triangular array A226314(n)/A054531(n), then a(n) selects the prime rationals p/q from the same array.

Links

Table of n, a(n) for n=1..58.

Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.

A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arithmetica IV., 1957.

Wikipedia Schinzel's hypothesis H.

Example

a(7)=57 because A226314(7)/A054531(7)=1/4 and with least x=3 we have p=f_1=x-1=2 and q=f_2=4x-1=11. Therefore (p+1)/(q+1)=3/12=1/4. Also A226314(57)/A054531(57)=p/q=2/11.

Mathematica

func[{i_, j_}] := {j(j-1)/2+i->{j+(i-j)/GCD[i, j], j/GCD[i, j]}}; rfunc[{i_, j_}] := {{j+(i-j)/GCD[i, j], j/GCD[i, j]}->j(j-1)/2+i}; getx[{a_, b_}] := Module[{f1, f2, x}, If[a==b, {1, 1}, (f1=a*x-1; f2=b*x-1; x=1; While[(!PrimeQ[f1]||!PrimeQ[f2])&&x<10^5, x++]; If[x==10^5, Abort[], {f1, f2}])]]; assoc=Association@Flatten[Table[func[{a, b}], {b, 1000}, {a, b}], 1]; rassoc=Association@Flatten[Table[rfunc[{a, b}], {b, 1000}, {a, b}], 1]; Table[rassoc[getx[assoc[n]]], {n, 1, 100}]

Crossrefs

Cf. A054531, A226314, A278635.

Sequence in context: A258088 A259263 A341039 * A119147 A370395 A226176

Adjacent sequences: A279366 A279367 A279368 * A279370 A279371 A279372

Keyword

nonn

Author

Frank M Jackson and Michael B Rees, Dec 10 2016

Status

approved