A252232 -id:A252232 - cp's OEIS frontend

A252582 Triangular array of values of Wieferich primes p in A252232 read by rows.

2, 2, 3, 3, 7, 13, 2, 3, 11, 157, 2, 5, 7, 19, 83, 2, 11, 17, 109, 239, 401, 5, 7, 11, 13, 83, 173, 1259, 2, 3, 5, 11, 31, 37, 61, 109, 2, 3, 5, 7, 43, 293, 317, 383, 1627, 2, 5, 7, 11, 31, 47, 89, 167, 523, 619

Offset: 1

Felix Fröhlich, Dec 18 2014

			Triangle starts
  2
  2  3
  3  7 13
  2  3 11 157
  2  5  7  19  83
  2 11 17 109 239 401
  5  7 11  13  83 173 1259
  2  3  5  11  31  37   61 109
  2  3  5   7  43 293  317 383 1627
  2  5  7  11  31  47   89 167  523 619

Cf. A252232.

Flatten@Table[k=2;While[Length[s=Select[Prime@Range@PrimePi[k-1],PowerMod[k,(#-1),#^2]==1&]]!=n,k=NextPrime@k];s,{n,8}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A255885 Smallest base b such that there exist exactly n Wieferich pseudoprimes (composites c satisfying b^(c-1) == 1 (mod c^2)) less than b.

Original entry on oeis.org

17, 65, 145, 485, 649, 1297, 577, 2024, 5185, 8182, 7057, 8749, 14401, 30753, 56449, 57601, 77401, 129473, 51841, 129601, 254017, 296449, 202501, 389377

Offset: 1

Felix Fröhlich, Mar 09 2015

Cf. A244752, A252232, A255901.

for(n=1, 10, b=2; while(b > 0, i=0; forcomposite(c=2, b, if(Mod(b, c^2)^(c-1)==1, i++)); if(i==n, print1(b, ", "); break({1})); b++))

from itertools import count
from sympy import isprime
def A255885(n):
    for b in count(1):
        if n == sum(1 for c in range(2,b+1) if not isprime(c) and pow(b,c-1,c**2) == 1):
            return b # Chai Wah Wu, May 18 2022

a(20) from Chai Wah Wu, May 18 2022

a(21)-a(24) from Chai Wah Wu, May 19 2022

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.

Original entry on oeis.org

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

Offset: 1

Felix Fröhlich, Mar 10 2015

			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)

Cf. A252232, A255885.

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 *)

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++))

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

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

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

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

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

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A252582 Triangular array of values of Wieferich primes p in A252232 read by rows.

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A255885 Smallest base b such that there exist exactly n Wieferich pseudoprimes (composites c satisfying b^(c-1) == 1 (mod c^2)) less than b.

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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.

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Mathematica

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Python

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