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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".

%I A273785 #12 Apr 21 2017 04:38:26
%S A273785 17,26,33,37,49,65,73,80,81,82,97,99,101,109,113,129,145,146,161,163,
%T A273785 168,170,177,181,182,193,197,199,201,209,217,224,225,226,239,241,242,
%U A273785 244,251,253,257,268,273,289,293,301,305,321,323,325,337,353,360,361
%N 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".
%C A273785 Contains n+1 for n in A048111. - _Robert Israel_, Apr 20 2017
%H A273785 Robert Israel, <a href="/A273785/b273785.txt">Table of n, a(n) for n = 1..5000</a>
%e A273785 15 satisfies the congruence 26^(15-1) == 1 (mod 15^2) and 15 < 26, so 26 is a term of the sequence.
%p A273785 N:= 1000: # to get all terms <= N
%p A273785 Res:= {}:
%p A273785 for c from 4 to N-1 do
%p A273785   if not isprime(c) then
%p A273785     for m in map(rhs@op, [msolve(x^(c-1)-1, c^2)]) do
%p A273785        if m > c and m <= N then Res:= Res union {m, seq(k*c^2+m, k=1..(N-m)/c^2)}
%p A273785        else Res:= Res union {seq(k*c^2+m, k=1..(N-m)/c^2)}
%p A273785        fi
%p A273785     od
%p A273785   fi
%p A273785 od:
%p A273785 sort(convert(Res,list)); # _Robert Israel_, Apr 20 2017
%t A273785 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 *)
%o A273785 (PARI) is(n) = forcomposite(c=1, n-1, if(Mod(n, c^2)^(c-1)==1, return(1))); return(0)
%Y A273785 Cf. A048111, A240719, A244752, A255885, A256517, A267288, A268310, A273339, A273340.
%K A273785 nonn
%O A273785 1,1
%A A273785 _Felix Fröhlich_, May 30 2016