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Smallest j such that 2*j*prime(n)^3-1 is prime.
5

%I #27 Sep 08 2022 08:46:04

%S 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 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 6,12,11,6,5,1,9,2,19,5,9,27,1,3,11,3,15,2,6,21

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

%C 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 Pierre CAMI, <a href="/A224609/b224609.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A053989(2p^3) where p is the n-th prime. - _Charles R Greathouse IV_, Apr 18 2013

%e 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 1*2*3^3-1=53 is prime, so a(2)=1 as p(2)=3.

%e 1*2*5^3-1=249 is composite; 2*2*5^3=499 is prime, so a(3)=2 as p(3)=5.

%t 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 (Magma)

%o S:=[];

%o j:=1;

%o for n in [1..100] do

%o while not IsPrime(2*j*NthPrime(n)^3-1) do

%o j:=j+1;

%o end while;

%o Append(~S, j);

%o j:=1;

%o end for;

%o S; // _Bruno Berselli_, Apr 18 2013

%o (PARI) a(n)=my(P=2*prime(n)^3,j);while(!isprime(j++*P-1),);j \\ _Charles R Greathouse IV_, Apr 18 2013

%Y Cf. A224489.

%K nonn

%O 1,1

%A _Pierre CAMI_, Apr 12 2013