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".

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

1,1

Comments

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

Links

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

Example

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A129910 A259075 A212909 * A171169 A316561 A034963

Adjacent sequences: A273782 A273783 A273784 * A273786 A273787 A273788

Keyword

nonn

Author

Felix Fröhlich, May 30 2016

Status

approved