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A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

%I A182297 #39 Aug 03 2014 14:37:24
%S A182297 21,39,55,57,105,111,147,155,165,171,183,195,201,203,205,219,231,237,
%T A182297 253,273,285,291,301,305,309,327,333,355,357,385,399,417,429,453,465,
%U A182297 483,489,495,497,505,507,525,543,555,579,597,605,609,615,627,633,651,655
%N A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.
%C A182297 The primes in this sequence are A001220, the Wieferich primes. - _Charles R Greathouse IV_, Feb 02 2014
%C A182297 Odd prime p is a Wieferich prime if and only if A002326((p^2-1)/2) = A002326((p-1)/2). See the sixth comment to A001220 and my formula below. - _Thomas Ordowski_, Feb 03 2014
%H A182297 Alois P. Heinz, <a href="/A182297/b182297.txt">Table of n, a(n) for n = 1..1000</a>
%H A182297 Z. Franco and C. Pomerance, <a href="http://dx.doi.org/10.1090/S0025-5718-1995-1297468-4">On a conjecture of Crandall concerning the qx + 1 problem</a>, Math. Comp. Vol. 64, No. 211 (1995), 1333-1336.
%F A182297 Odd numbers q such that A002326((q^2-1)/2) < q * A002326((q-1)/2). Other positive odd integers satisfy the equality. - _Thomas Ordowski_, Feb 03 2014
%F A182297 Odd numbers q such that gcd(A165781((q-1)/2), q) > 1. - _Thomas Ordowski_, Feb 12 2014
%e A182297 21 is in the sequence because the multiplicative order of 2 mod 21 is 6, and (2^6-1)/21 = 3, which is not coprime to 21.
%p A182297 with(numtheory):
%p A182297 a:= proc(n) option remember; local q;
%p A182297       for q from 2 +`if`(n=1, 1, a(n-1)) by 2
%p A182297         while igcd((2^order(2, q)-1)/q, q)=1 do od; q
%p A182297     end:
%p A182297 seq (a(n), n=1..60);  # _Alois P. Heinz_, Apr 23 2012
%t A182297 Select[Range[1, 799, 2], GCD[#, (2^MultiplicativeOrder[2, #] - 1)/#] > 1 &] (* _Alonso del Arte_, Apr 23 2012 *)
%o A182297 (PARI) is(n)=n%2 && gcd(lift(Mod(2,n^2)^znorder(Mod(2,n))-1)/n,n)>1 \\ _Charles R Greathouse IV_, Feb 02 2014
%Y A182297 For another definition of Wieferich numbers, see A077816.
%Y A182297 Cf. A002326.
%K A182297 nonn
%O A182297 1,1
%A A182297 _Felix Fröhlich_, Apr 23 2012