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A255901 Smallest base b such that there exist exactly n Wieferich primes (primes p satisfying b^(p-1) == 1 (mod p^2)) less than b.

%I A255901 #25 Nov 13 2024 05:38:11
%S A255901 5,17,19,116,99,361,1451,1693,10768,13834,208301,548291
%N A255901 Smallest base b such that there exist exactly n Wieferich primes (primes p satisfying b^(p-1) == 1 (mod p^2)) less than b.
%F A255901 For all n a(n) <= A252232(n).
%F A255901 a(n) = A252232(n) iff a(n) is prime.
%e A255901 From _Robert G. Wilson v_, Mar 11 2015: (Start)
%e A255901 n        b  p
%e A255901 1:       5 {2}
%e A255901 2:      17 {2, 3}
%e A255901 3:      19 {3, 7, 13}
%e A255901 4:     116 {3, 7, 19, 47}
%e A255901 5:      99 {5, 7, 13, 19, 83}
%e A255901 6:     361 {2, 3, 7, 13, 43, 137}
%e A255901 7:    1451 {5, 7, 11, 13, 83, 173, 1259}
%e A255901 8:    1693 {2, 3, 5, 11, 31, 37, 61, 109}
%e A255901 9:   10768 {5, 11, 17, 19, 79, 101, 139, 6343, 10177}
%e A255901 10:  13834 {3, 11, 17, 19, 43, 139, 197, 2437, 5849, 6367}
%e A255901 11: 208301 {2, 5, 29, 47, 59, 113, 661, 8209, 13679, 15679, 55633}
%e A255901 12: 548291 {7, 11, 19, 29, 31, 37, 97, 211, 547, 911, 2069, 28927}
%e A255901 ... (End)
%t A255901 f[n_] := Block[{b = 2, p}, While[p = Prime@ Range@ PrimePi[b - 1]; Count[ PowerMod[b, p - 1, p^2], 1] != n, b++]; b]; Array[f, 11] (* _Robert G. Wilson v_, Mar 11 2015 *)
%o A255901 (PARI) for(n=1, 10, b=2; while(b > 0, i=0; forprime(p=1, b, if(Mod(b, p^2)^(p-1)==1, i++)); if(i==n, print1(b, ", "); break({1})); b++))
%o A255901 (Python)
%o A255901 from itertools import count
%o A255901 from sympy import primerange
%o A255901 def A255901(n):
%o A255901     for b in count(1):
%o A255901         if n == sum(1 for p in primerange(2,b+1) if pow(b,p-1,p**2) == 1):
%o A255901             return b # _Chai Wah Wu_, May 18 2022
%Y A255901 Cf. A252232, A255885.
%K A255901 nonn,more
%O A255901 1,1
%A A255901 _Felix Fröhlich_, Mar 10 2015
%E A255901 a(11) from _Robert G. Wilson v_, Mar 11 2015
%E A255901 a(12) from _Robert G. Wilson v_, Mar 12 2015