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

1,1

Links

Table of n, a(n) for n=1..12.

Formula

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

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

Example

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)

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 *)

Prog

(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

Crossrefs

Cf. A252232, A255885.

Sequence in context: A171255 A304540 A306125 * A098333 A252232 A162862

Adjacent sequences: A255898 A255899 A255900 * A255902 A255903 A255904

Keyword

nonn,more

Author

Felix Fröhlich, Mar 10 2015

Extensions

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

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

Status

approved