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

1,2

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).

Links

Table of n, a(n) for n=1..89.

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)=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.

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}]

Crossrefs

Cf. A226314, A054531.

Sequence in context: A332518 A353662 A062366 * A294656 A057937 A080216

Adjacent sequences: A278632 A278633 A278634 * A278636 A278637 A278638

Keyword

nonn

Author

Frank M Jackson and Michael B Rees, Nov 24 2016

Status

approved