cp's OEIS Frontend

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).

%I A278635 #7 Jun 27 2023 15:07:43
%S A278635 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,
%T A278635 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,
%U A278635 4,2,2,2,4,4,2,4,12,4,4,4,8,24,8,8,18,8,14,24,8,8,18
%N 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).
%C A278635 Rationals a/b (lowest form) can be mapped uniquely into a triangular array A226314(n)/A054531(n).
%C A278635 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).
%H A278635 Lance Fortnow, <a href="http://blog.computationalcomplexity.org/2004/03/counting-rationals-quickly.html">Counting the Rationals Quickly</a>, Computational Complexity Weblog, Monday, March 01, 2004.
%H A278635 A. Schinzel and W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf">Sur certaines hypothèses concernant les nombres premiers</a>, Acta Arithmetica IV., 1957.
%H A278635 Wikipedia <a href="http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's hypothesis H</a>.
%e A278635 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.
%t A278635 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}]
%Y A278635 Cf. A226314, A054531.
%K A278635 nonn
%O A278635 1,2
%A A278635 _Frank M Jackson_ and Michael B Rees, Nov 24 2016