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

2, 3, 7, 43, 3613, 65250781, 5109197227031017, 21753246920584523633819544186061, 993727878334632126576336773629979379563850938567846991629270287

Offset: 1

Author

Rick L. Shepherd, May 31 2002

Keywords

• nonn

Comments

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

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

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

Links

• Joerg Arndt, Table of n, a(n) for n = 1..13
• Mersenne Forum, A071580

Crossrefs

Cf. A061092, A258081.

Programs

Maple

P:= 1:
for n from 1 to 13 do
  for k from 1 do
     if isprime(k*P+1) then
       A[n]:= k*P+1;
       P:= P * A[n];
       break
     fi
  od
od:
seq(A[i],i=1..13); # Robert Israel, May 19 2015

Mathematica

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 *)

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

Extensions

Definition reworded by Andrew R. Booker, May 19 2015