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%I #23 Dec 29 2014 18:37:19
%S 1093,11,2,5,71,2,2,3,13,2,7,2,2,5
%N 1st Wieferich prime base prime(n).
%C Smallest prime p such that p^2 divides prime(n)^(p-1) - 1.
%C Smallest prime p such that p divides the Fermat quotient q_p((prime(n)) = (prime(n)^(p-1) - 1)/p.
%C See additional comments, links, and cross-refs in A039951.
%C a(15) = A039951(47) > 4.1*10^13.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_quotient#Generalized_Wieferich_primes"> Generalized Wieferich primes</a>.
%H Robert G. Wilson v, <a href="/A174422/a174422.txt">Table of n, a(n) for n = 1..10000 (or 0 if unknown)</a>
%F a(n) = A039951(prime(n)).
%F a(n) = 2 if and only if prime(n) == 1 (mod 4). [_Jonathan Sondow_, Aug 29 2010]
%e a(1) = 1093 is the first Wieferich prime A001220. a(2) = 11 is the first Mirimanoff prime A014127.
%t f[n_] := Block[{b = Prime@ n, p = 2}, While[ PowerMod[b, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 14]
%o (PARI) forprime(a=2, 20, forprime(p=2, 10^9, if(Mod(a, p^2)^(p-1)==1, print1(p, ", "); next({2}))); print1("--, ")) \\ _Felix Fröhlich_, Jun 27 2014
%Y Cf. A001220, A014127, A039951 = smallest prime p such that p^2 divides n^(p-1) - 1, A125636 = smallest prime p such that prime(n)^2 divides p^(prime(n)-1) - 1.
%Y Cf. A178871 = 2nd Wieferich prime base prime(n).
%K hard,more,nonn
%O 1,1
%A _Jonathan Sondow_, Mar 19 2010
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