A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)*...*a(n-1)*p).

3, 5, 13, 37, 73, 109, 181, 541, 1621, 4861, 9721, 19441, 58321, 87481, 379081, 408241, 2041201, 2449441, 7348321, 14696641, 22044961, 95528161, 382112641, 2292675841, 8024365441, 40121827201, 481461926401, 722192889601, 2888771558401, 7944121785601, 55608852499201, 111217704998401, 889741639987201, 1779483279974401

Offset: 1

Author

Manuel Valdivia, Mar 15 2010

Keywords

• nonn

Comments

Conjecture: a(n) is the smallest integer k > 1 such that 2^(k-1) == 1 (mod a(0)*...*a(n-1)*k), with a(0) = 1. - Thomas Ordowski, Mar 13 2019

Either a(n) > a(n-1), or a(n) = a(n-1) is a Wieferich prime (A001220). - Max Alekseyev, Sep 29 2024

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007663, A096060, A002200, A005109, A306826.

Programs

Mathematica

i=1;Do[p=Prime[n];If[Mod[2^(p-1)-1,p*i]==0,Print[p];i=p*i],{n,2,78498}]

PARI

findprime(prd) = {forprime(p=2, , if (Mod(2, p*prd)^(p-1) == 1, return (p)););}
lista(nn) = {my(prd = 1, na); for (n=1, nn, na = findprime(prd); print1(na, ", "); prd *= na;);} \\ Michel Marcus, Mar 14 2019

PARI

{ a175257_first_terms(N=1000) = my(P,L,t); P=[3]; L=2; for(n=#P,N, print(n," ",P[n]); forstep(p=P[n],oo,Mod(1,L), if(p==P[n], if(Mod(2,p^2)^(p-1)==1, error("Wieferich prime!"), next)); if(ispseudoprime(p), P=concat(P,[p]); t=Mod(2,p)^L; fordiv((p-1)\L,d, if(t^d==1, L*=d; break)); break))); P; } \\ Max Alekseyev, Sep 29 2024

Extensions

a(17)-a(26) from Amiram Eldar, Feb 03 2019

Name corrected by Thomas Ordowski, Mar 13 2019

a(27) from Hans Havermann, Mar 29 2019

Eliminated a(0)=1 in the definition (empty products equal 1). - R. J. Mathar, Jun 19 2021

Terms a(28) onward from Max Alekseyev, Sep 29 2024