A255885 -id:A255885 - cp's OEIS frontend

A256517 Let c be the n-th composite number. Then a(n) is the smallest base b > 1 such that b^(c-1) == 1 (mod c^2), i.e., such that c is a 'Wieferich pseudoprime'.

17, 37, 65, 80, 101, 145, 197, 26, 257, 325, 401, 197, 485, 577, 182, 677, 728, 177, 901, 1025, 485, 1157, 99, 1297, 1445, 170, 1601, 1765, 1937, 82, 2117, 2305, 1047, 2501, 577, 529, 2917, 1451, 3137, 721, 3365, 3601, 3845, 244, 4097, 99, 1945, 4625, 530

Offset: 1

Felix Fröhlich, Apr 01 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Felix Fröhlich)

Cf. A039678, A242742, A244752, A255885.

Cf. A185103, A002808.

c = Select[Range@ 69, CompositeQ]; f[c_] := Block[{b = 2}, While[Mod[b^(c - 1), c^2] != 1, b++]; b]; f /@ c (* Michael De Vlieger, Apr 03 2015 *)

forcomposite(c=1, 1e3, b=2; while(Mod(b, c^2)^(c-1)!=1, b++); print1(b, ", "))

from sympy import composite
from sympy.ntheory.residue_ntheory import nthroot_mod
def A256517(n):
    z = nthroot_mod(1,(c := composite(n))-1,c**2,True)
    return int(z[0]+c**2 if len(z) == 1 else z[1]) # Chai Wah Wu, May 18 2022

a(n) = A185103(A002808(n)-1). - Bill McEachen, Nov 27 2021

A267288 Composites c where at least one base b with 1 < b < c exists such that b^(c-1) == 1 (mod c^2), i.e., composites c that are base-b 'Wieferich pseudoprimes' for at least one b between 1 and c.

Original entry on oeis.org

133, 451, 561, 871, 904, 1065, 1105, 1267, 1729, 1891, 2059, 2201, 2501, 2821, 2993, 3145, 4641, 5005, 5551, 5841, 5963, 6409, 6541, 6601, 6697, 7107, 7471, 7501, 8321, 8323, 9637, 10585, 11266, 12209, 12403, 13571, 13585, 16471, 17261, 17466, 17770, 18103

Offset: 1

Felix Fröhlich, Jan 12 2016

nonn

A002808(i) such that A256517(i) < A002808(i).

Any term is also a term of A039769.

			871 is a term of the sequence, since 699^870 == 1 (mod 871^2) and 699 < 871.

Cf. A039769, A134307, A255885, A256517.

forcomposite(c=2, , for(b=2, c-1, if(Mod(b, c^2)^(c-1)==1, print1(c, ", "); break({1}))))

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

A273785 Numbers n where a composite c < n exists such that n^(c-1) == 1 (mod c^2), i.e., such that c is a "base-n Wieferich pseudoprime".

Original entry on oeis.org

17, 26, 33, 37, 49, 65, 73, 80, 81, 82, 97, 99, 101, 109, 113, 129, 145, 146, 161, 163, 168, 170, 177, 181, 182, 193, 197, 199, 201, 209, 217, 224, 225, 226, 239, 241, 242, 244, 251, 253, 257, 268, 273, 289, 293, 301, 305, 321, 323, 325, 337, 353, 360, 361

Offset: 1

Felix Fröhlich, May 30 2016

nonn

Contains n+1 for n in A048111. - Robert Israel, Apr 20 2017

			15 satisfies the congruence 26^(15-1) == 1 (mod 15^2) and 15 < 26, so 26 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A048111, A240719, A244752, A255885, A256517, A267288, A268310, A273339, A273340.

N:= 1000: # to get all terms <= N
Res:= {}:
for c from 4 to N-1 do
  if not isprime(c) then
    for m in map(rhs@op, [msolve(x^(c-1)-1, c^2)]) do
       if m > c and m <= N then Res:= Res union {m, seq(k*c^2+m, k=1..(N-m)/c^2)}
       else Res:= Res union {seq(k*c^2+m, k=1..(N-m)/c^2)}
       fi
    od
  fi
od:
sort(convert(Res,list)); # Robert Israel, Apr 20 2017

nn = 361; c = Select[Range@ nn, CompositeQ]; Select[Range@ nn, Function[n, Count[TakeWhile[c, # <= n &], k_ /; Mod[n^(k - 1), k^2] == 1] > 0]] (* Michael De Vlieger, May 30 2016 *)

is(n) = forcomposite(c=1, n-1, if(Mod(n, c^2)^(c-1)==1, return(1))); return(0)

cp's OEIS Frontend

A256517 Let c be the n-th composite number. Then a(n) is the smallest base b > 1 such that b^(c-1) == 1 (mod c^2), i.e., such that c is a 'Wieferich pseudoprime'.

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A267288 Composites c where at least one base b with 1 < b < c exists such that b^(c-1) == 1 (mod c^2), i.e., composites c that are base-b 'Wieferich pseudoprimes' for at least one b between 1 and c.

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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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A273785 Numbers n where a composite c < n exists such that n^(c-1) == 1 (mod c^2), i.e., such that c is a "base-n Wieferich pseudoprime".

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