A255901 - OEIS

%I #25 Nov 13 2024 05:38:11

%S 5,17,19,116,99,361,1451,1693,10768,13834,208301,548291

%N 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 For all n a(n) <= A252232(n).

%F a(n) = A252232(n) iff a(n) is prime.

%e From _Robert G. Wilson v_, Mar 11 2015: (Start)

%e n        b  p

%e 1:       5 {2}

%e 2:      17 {2, 3}

%e 3:      19 {3, 7, 13}

%e 4:     116 {3, 7, 19, 47}

%e 5:      99 {5, 7, 13, 19, 83}

%e 6:     361 {2, 3, 7, 13, 43, 137}

%e 7:    1451 {5, 7, 11, 13, 83, 173, 1259}

%e 8:    1693 {2, 3, 5, 11, 31, 37, 61, 109}

%e 9:   10768 {5, 11, 17, 19, 79, 101, 139, 6343, 10177}

%e 10:  13834 {3, 11, 17, 19, 43, 139, 197, 2437, 5849, 6367}

%e 11: 208301 {2, 5, 29, 47, 59, 113, 661, 8209, 13679, 15679, 55633}

%e 12: 548291 {7, 11, 19, 29, 31, 37, 97, 211, 547, 911, 2069, 28927}

%e ... (End)

%t 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 (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 (Python)

%o from itertools import count

%o from sympy import primerange

%o def A255901(n):

%o     for b in count(1):

%o         if n == sum(1 for p in primerange(2,b+1) if pow(b,p-1,p**2) == 1):

%o             return b # _Chai Wah Wu_, May 18 2022

%Y Cf. A252232, A255885.

%K nonn,more

%O 1,1

%A _Felix Fröhlich_, Mar 10 2015

%E a(11) from _Robert G. Wilson v_, Mar 11 2015

%E a(12) from _Robert G. Wilson v_, Mar 12 2015