A255901 -id:A255901 - cp's OEIS frontend

A256236 Smallest b > 1 such that the first n primes p (i.e., A000040(1)-A000040(n)) all satisfy b^(p-1) == 1 (mod p^2), i.e., smallest base b larger than 1 such that any member of the set of first n primes is a base-b Wieferich prime.

5, 17, 449, 557, 19601, 132857, 4486949, 126664001, 2363321449, 5229752849, 2486195039249, 16250570614349, 83322586961893, 39699586259362801, 8042447016668335049, 449320365877347849601, 4376479338174582826793

Offset: 1

Felix Fröhlich, Mar 25 2015

There might be bases b where prime(n+1) is also a base-b Wieferich prime. This does not affect the membership of b in the sequence.
Are there any terms such that a(n) = a(n+1)?
Does b exist for all n?
All currently known terms satisfy a(n) >= A255901(n). Are there any terms such that a(n) < A255901(n)?
If it exists, a(12) > 6*10^12. - Robert Price, Oct 10 2019
a(n) <= prime(n)#^2+1 = A189409(n), since any prime p is a Wieferich prime in base k*p^2+1 for all k. - Jens Kruse Andersen, Dec 20 2020

			Values of bases b and the values of first Wieferich primes p to base b:
b             | p
-------------------------------------------------------------------------
5             | 2, 20771, 40487 ...
17            | 2, 3, 46021, 48947 ...
449           | 2, 3, 5, 1789 ...
557           | 2, 3, 5, 7, 23, 39829 ...
19601         | 2, 3, 5, 7, 11, 23, 47 ...
132857        | 2, 3, 5, 7, 11, 13, 73, 257 ...
4486949       | 2, 3, 5, 7, 11, 13, 17, 89, 197 ...
126664001     | 2, 3, 5, 7, 11, 13, 17, 19, 101, 2789 ...
2363321449    | 2, 3, 5, 7, 11, 13, 17, 19, 23 ...
5229752849    | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 881, 2246969 ...
2486195039249 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 ...

Cf. A255901.

b = 2; Table[While[fnd = True;
  For[i = 1, i <= n, i++,
   p = Prime[i];
   If[PowerMod[b, (p - 1), p^2] != 1 , fnd = False;  Break[]]];
b++; ! fnd]; b - 1, {n, 5}] (* Robert Price, Oct 10 2019 *)

a(n) = my(v=primes(n)); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

a(9)-a(11) from Robert Price, Oct 10 2019

a(12)-a(17) from Jens Kruse Andersen, Dec 28 2020

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

cp's OEIS Frontend

A256236 Smallest b > 1 such that the first n primes p (i.e., A000040(1)-A000040(n)) all satisfy b^(p-1) == 1 (mod p^2), i.e., smallest base b larger than 1 such that any member of the set of first n primes is a base-b Wieferich prime.

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