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A224609 Smallest j such that 2*j*prime(n)^3-1 is prime.

%I A224609 #27 Sep 08 2022 08:46:04
%S A224609 2,1,2,7,2,7,8,6,8,5,1,3,11,1,9,3,5,1,3,15,7,3,8,8,12,2,15,3,10,2,3,
%T A224609 12,12,1,6,6,9,3,5,2,5,1,5,10,57,1,21,1,15,9,2,3,1,5,5,3,15,6,7,5,25,
%U A224609 6,12,11,6,5,1,9,2,19,5,9,27,1,3,11,3,15,2,6,21
%N A224609 Smallest j such that 2*j*prime(n)^3-1 is prime.
%C A224609 We are searching smallest j such that j*prime(n)*2*p(n)^2-1 is prime, for A224489 it is smallest k such that k*2*prime(n)^2-1 is prime, so here we replace smallest k by smallest j*prime(n).
%H A224609 Pierre CAMI, <a href="/A224609/b224609.txt">Table of n, a(n) for n = 1..10000</a>
%F A224609 a(n) = A053989(2p^3) where p is the n-th prime. - _Charles R Greathouse IV_, Apr 18 2013
%e A224609 1*2*2^3-1= 15 is composite; 2*2*2^3-1= 31 is prime, so a(1)=2 as p(1)=2.
%e A224609 1*2*3^3-1=53 is prime, so a(2)=1 as p(2)=3.
%e A224609 1*2*5^3-1=249 is composite; 2*2*5^3=499 is prime, so a(3)=2 as p(3)=5.
%t A224609 jmax = 10^5 (* sufficient up to 10^5 terms *); a[n_] := For[j = 1, j <= jmax, j++, p = Prime[n]; If[PrimeQ[j*2*p^3 - 1], Return[j]]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Apr 18 2013 *)
%o A224609 (Magma)
%o A224609 S:=[];
%o A224609 j:=1;
%o A224609 for n in [1..100] do
%o A224609   while not IsPrime(2*j*NthPrime(n)^3-1) do
%o A224609        j:=j+1;
%o A224609   end while;
%o A224609   Append(~S, j);
%o A224609   j:=1;
%o A224609 end for;
%o A224609 S; // _Bruno Berselli_, Apr 18 2013
%o A224609 (PARI) a(n)=my(P=2*prime(n)^3,j);while(!isprime(j++*P-1),);j \\ _Charles R Greathouse IV_, Apr 18 2013
%Y A224609 Cf. A224489.
%K A224609 nonn
%O A224609 1,1
%A A224609 _Pierre CAMI_, Apr 12 2013