cp's OEIS Frontend

A071580 Smallest prime of the form k*a(n-1)*a(n-2)*...*a(1)+1.

%I A071580 #27 May 14 2023 09:39:23
%S A071580 2,3,7,43,3613,65250781,5109197227031017,
%T A071580 21753246920584523633819544186061,
%U A071580 993727878334632126576336773629979379563850938567846991629270287
%N A071580 Smallest prime of the form k*a(n-1)*a(n-2)*...*a(1)+1.
%C A071580 The former definition was "Smallest prime == 1 mod (a(n-1)*a(n-2)*...*a(1)) for n>=2 with a(1)=2."
%C A071580 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 A071580 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 A071580 Joerg Arndt, <a href="/A071580/b071580.txt">Table of n, a(n) for n = 1..13</a>
%H A071580 Mersenne Forum, <a href="http://mersenneforum.org/showthread.php?t=20260">A071580</a>
%p A071580 P:= 1:
%p A071580 for n from 1 to 13 do
%p A071580   for k from 1 do
%p A071580      if isprime(k*P+1) then
%p A071580        A[n]:= k*P+1;
%p A071580        P:= P * A[n];
%p A071580        break
%p A071580      fi
%p A071580   od
%p A071580 od:
%p A071580 seq(A[i],i=1..13); # _Robert Israel_, May 19 2015
%t A071580 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 A071580 (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 A071580 Cf. A061092, A258081.
%K A071580 nonn
%O A071580 1,1
%A A071580 _Rick L. Shepherd_, May 31 2002
%E A071580 Definition reworded by _Andrew R. Booker_, May 19 2015