cp's OEIS Frontend

A125645 Smallest odd prime base q such that p^4 divides q^(p-1) - 1, where p = prime(n).

%I A125645 #18 Mar 20 2025 22:33:23
%S A125645 17,163,443,3449,45989,239,15541,2819,60793,78017,690143,398023,
%T A125645 1977343,574081,1513367,4388179,3198427,8065789,3246107,1353383,
%U A125645 5934307,15631613,2864371,14754769,15012733,1358891,32414783,119551,21860063,11281097
%N A125645 Smallest odd prime base q such that p^4 divides q^(p-1) - 1, where p = prime(n).
%H A125645 Robert Israel, <a href="/A125645/b125645.txt">Table of n, a(n) for n = 1..10000</a>
%H A125645 W. Keller and J. Richstein <a href="http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients that are divisible by p</a>.
%p A125645 f:= proc(n) local p, r, S,i,s,t;
%p A125645   uses numtheory;
%p A125645   p:= ithprime(n);
%p A125645   r:= primroot(p^4);
%p A125645   S:= sort([seq(r &^ (i*p^3) mod p^4, i=0..p-2)]);
%p A125645   for i from 0 do
%p A125645     for s in S do
%p A125645       t:= i*p^4+s;
%p A125645       if t::odd and isprime(t) then return t fi
%p A125645   od od
%p A125645 end proc:
%p A125645 f(1):= 1:
%p A125645 map(f, [$1..100]); # _Robert Israel_, Feb 12 2017
%o A125645 (PARI) { a(n) = local(p,x,y); if(n==1,return(17)); p=prime(n); x=znprimroot(p^4)^(p^3); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^4);y ) )[1] } \\ _Max Alekseyev_, May 30 2007
%Y A125645 Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125646, A125647, A125648, A125649.
%K A125645 nonn
%O A125645 1,1
%A A125645 _Alexander Adamchuk_, Nov 29 2006
%E A125645 More terms from _Max Alekseyev_, May 30 2007