A278635 Least x such that ax-1 and bx-1 are simultaneously prime and n (sequence index) is the position of rational a/b (lowest form) in the triangular array A226314(n)/A054531(n).

1, 3, 3, 4, 2, 4, 3, 2, 1, 3, 4, 4, 4, 6, 4, 3, 1, 4, 4, 4, 3, 6, 2, 2, 2, 6, 2, 6, 3, 6, 1, 2, 4, 2, 6, 3, 6, 2, 2, 2, 6, 1, 2, 6, 6, 3, 4, 2, 6, 2, 4, 2, 6, 2, 3, 4, 4, 4, 12, 4, 4, 12, 4, 10, 18, 4, 4, 2, 2, 2, 4, 4, 2, 4, 12, 4, 4, 4, 8, 24, 8, 8, 18, 8, 14, 24, 8, 8, 18

Offset: 1

Author

Frank M Jackson and Michael B Rees, Nov 24 2016

Keywords

• nonn

Comments

Rationals a/b (lowest form) can be mapped uniquely into a triangular array A226314(n)/A054531(n).

By Schinzel's Hypothesis H the functions f_1=ax-1 and f_2=bx-1 have infinite values of x such that f_1 and f_2 are simultaneously prime. Hence a/b can be expressed using two primes p and q as a/b=(p+1)/(q+1). This sequence determines the least x for generating p=f_1 and q=f_2 with the sequence index n selecting a/b from the triangular array A226314(n)/A054531(n).

Examples

			a(7)=3 because A226314(7)/A054531(7)=1/4 and with 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.
		

Links

• 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.

Crossrefs

Cf. A226314, A054531.

Programs

Mathematica

func[{i_, j_}] := {j(j-1)/2+i->{j+(i-j)/GCD[i, j], j/GCD[i, j]}}; getx[{a_, b_}] := Module[{f1, f2, x}, If[a==b, 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[], x])]]; assoc=Association@Flatten[Table[func[{a, b}], {b, 1000}, {a, b}], 1]; Table[getx[assoc[n]], {n, 1, 100}]