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Smallest prime of the form k*a(n-1)*a(n-2)*...*a(1)+1.
5

%I #27 May 14 2023 09:39:23

%S 2,3,7,43,3613,65250781,5109197227031017,

%T 21753246920584523633819544186061,

%U 993727878334632126576336773629979379563850938567846991629270287

%N Smallest prime of the form k*a(n-1)*a(n-2)*...*a(1)+1.

%C The former definition was "Smallest prime == 1 mod (a(n-1)*a(n-2)*...*a(1)) for n>=2 with a(1)=2."

%C a(6) through a(13), with digit lengths 8, 16, 32, 63, 127, 253, 507 and 1012, respectively, have been certified prime with Primo.

%C There is no need to use Elliptic curve primality proving (ECPP) to certify the primes. The primality of each term can be proved recursively with the "N-1 method" since the full factorization of a(n)-1 is known. - _Jeppe Stig Nielsen_, May 14 2023

%H Joerg Arndt, <a href="/A071580/b071580.txt">Table of n, a(n) for n = 1..13</a>

%H Mersenne Forum, <a href="http://mersenneforum.org/showthread.php?t=20260">A071580</a>

%p P:= 1:

%p for n from 1 to 13 do

%p for k from 1 do

%p if isprime(k*P+1) then

%p A[n]:= k*P+1;

%p P:= P * A[n];

%p break

%p fi

%p od

%p od:

%p seq(A[i],i=1..13); # _Robert Israel_, May 19 2015

%t sp[{p_,a_}]:=Module[{k=1},While[!PrimeQ[k*p+1],k++];{p(p*k+1),p*k+1}]; NestList[sp,{2,2},10][[All,2]] (* _Harvey P. Dale_, Mar 04 2019 *)

%o (PARI) terms=13; v=vector(terms); p=2; v[1]=p; for(n=2,terms, q=p+1; while(!isprime(q), q=q+p); v[n]=q; p=p*q); v

%Y Cf. A061092, A258081.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, May 31 2002

%E Definition reworded by _Andrew R. Booker_, May 19 2015