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

Author

Felix Fröhlich, May 30 2016

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

PARI

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