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Smallest prime number P such that 2*P^(2*n)-1 is a prime number.
1

%I #26 Apr 25 2016 11:50:02

%S 2,2,2,3,13,2,7,2,2,3,79,5,83,1427,2,83,47,7,31,139,23,53,7,373,193,

%T 71,5,3,463,2,3,13,281,3,281,59,17,13,1399,7,61,163,151,2,103,479,89,

%U 5,127,421,457,857,2,43,101,349,359,2243,13,13,17,1451,2

%N Smallest prime number P such that 2*P^(2*n)-1 is a prime number.

%C a(n) = 2 when 2*n + 1 is a prime Mersenne exponent (A000043).

%H Pierre CAMI, <a href="/A240166/b240166.txt">Table of n, a(n) for n = 1..700</a>

%e 2*2^(2*1) - 1 = 7 prime so a(1) = 2.

%e 2*2^(2*2) - 1 = 31 prime so a(2) = 2.

%e 2*2^(2*3) - 1 = 127 prime so a(3) = 2.

%p A:= proc(n)

%p local p;

%p p:= 2;

%p while not isprime(2*p^(2*n)-1) do p:= nextprime(p) od:

%p p;

%p end proc:

%p seq(A(n),n=1..100); # _Robert Israel_, Sep 14 2014

%t a240166[n_Integer] := Module[{p = 1},

%t While[! PrimeQ[2*Prime[p]^(2*n) - 1], p++]; Prime[p]]; a240166 /@ Range[120] (* _Michael De Vlieger_, Aug 12 2014 *)

%o (PFGW & SCRIPT)

%o SCRIPT

%o DIM i

%o DIM n,0

%o DIM k

%o DIMS t

%o OPENFILEOUT myf,a(n).txt

%o LABEL loop1

%o SET n,n+1

%o SET i,0

%o LABEL loop2

%o SET i,i+1

%o SETS t,%d,%d\ ;n;p(i)

%o PRP 2*p(i)^(2*n)-1,t

%o IF ISPRP THEN GOTO a

%o GOTO loop2

%o LABEL a

%o WRITE myf,t

%o GOTO loop1

%o (PARI) a(n) = p=2; while(!ispseudoprime(2*p^(2*n)-1), p=nextprime(p+1)); p

%o vector(70, n, a(n)) \\ _Colin Barker_, Aug 02 2014

%Y Cf. A000043.

%K nonn

%O 1,1

%A _Pierre CAMI_, Aug 02 2014