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A252232 a(n) = smallest prime q where exactly n primes p exist such that p < q and q^(p-1) == 1 (mod p^2), i.e., smallest prime base q having exactly n Wieferich primes less than q.

%I A252232 #20 Mar 13 2015 15:19:21
%S A252232 5,17,19,233,293,977,1451,1693,33301,308093
%N A252232 a(n) = smallest prime q where exactly n primes p exist such that p < q and q^(p-1) == 1 (mod p^2), i.e., smallest prime base q having exactly n Wieferich primes less than q.
%C A252232 From _Robert G. Wilson v_, Mar 11 2015: (Start)
%C A252232 n        b  p
%C A252232 1:       5 {2}
%C A252232 2:      17 {2, 3}
%C A252232 3:      19 {3, 7, 13}
%C A252232 4:     233 {2, 3, 11, 157}
%C A252232 5:     293 {2, 5, 7, 19, 83}
%C A252232 6:     977 {2, 11, 17, 109, 239, 401}
%C A252232 7:    1451 {5, 7, 11, 13, 83, 173, 1259}
%C A252232 8:    1693 {2, 3, 5, 11, 31, 37, 61, 109}
%C A252232 9:   33301 {2, 3, 5, 7, 43, 293, 317, 383, 1627}
%C A252232 10: 308093 {2, 5, 7, 11, 31, 47, 89, 167, 523, 619}
%C A252232 ... (End)
%e A252232 a(5) = 293, because q = 293 is the smallest prime for which there are exactly five primes p with p < q such that q^(p-1) == 1 (mod p^2), namely 2, 5, 7, 19 and 83.
%o A252232 (PARI) for(n=1, 10, q=2; while(q > 1, q=nextprime(q+1); i=0; forprime(p=2, q, if(Mod(q, p^2)^(p-1)==1, i++); if(i==n, print1(q, ", "); break({2})))))
%Y A252232 Cf. A175932, A222206.
%Y A252232 For the values of p, see A252582.
%K A252232 nonn,hard,more
%O A252232 1,1
%A A252232 _Felix Fröhlich_, Dec 15 2014