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.

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

Offset: 1

Author

Felix Fröhlich, Mar 10 2015

Keywords

• nonn
• more

Examples

			From _Robert G. Wilson v_, Mar 11 2015: (Start)
n        b  p
1:       5 {2}
2:      17 {2, 3}
3:      19 {3, 7, 13}
4:     116 {3, 7, 19, 47}
5:      99 {5, 7, 13, 19, 83}
6:     361 {2, 3, 7, 13, 43, 137}
7:    1451 {5, 7, 11, 13, 83, 173, 1259}
8:    1693 {2, 3, 5, 11, 31, 37, 61, 109}
9:   10768 {5, 11, 17, 19, 79, 101, 139, 6343, 10177}
10:  13834 {3, 11, 17, 19, 43, 139, 197, 2437, 5849, 6367}
11: 208301 {2, 5, 29, 47, 59, 113, 661, 8209, 13679, 15679, 55633}
12: 548291 {7, 11, 19, 29, 31, 37, 97, 211, 547, 911, 2069, 28927}
... (End)
		

Crossrefs

Cf. A252232, A255885.

Programs

Mathematica

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 *)

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++))

Python

from itertools import count
from sympy import primerange
def A255901(n):
    for b in count(1):
        if n == sum(1 for p in primerange(2,b+1) if pow(b,p-1,p**2) == 1):
            return b # Chai Wah Wu, May 18 2022

Formula

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

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

Extensions

a(11) from Robert G. Wilson v, Mar 11 2015

a(12) from Robert G. Wilson v, Mar 12 2015