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Odd numbers m such that the order of 2 mod m^3 is less than m times the order of 2 mod m^2.
1

%I #13 Dec 10 2015 08:55:12

%S 57,111,219,285,327,399,489,505,543,555,597,627,741,777,813,969,1083,

%T 1095,1137,1221,1255,1299,1311,1379,1425,1443,1461,1467,1515,1533,

%U 1569,1623,1635,1653,1731,1767,1839,1887,1893,1995,2005,2109,2271,2289,2337,2409,2433,2445,2451,2487,2553,2649,2679,2715,2757,2775,2793,2811,2847,2973,2985,3005,3021,3027,3135,3189,3219,3351,3363,3423,3437,3441,3459,3477,3505,3513

%N Odd numbers m such that the order of 2 mod m^3 is less than m times the order of 2 mod m^2.

%C These numbers m are a subset of the {A182297} Wieferich numbers (2).

%C All known numbers m are composite. A prime p satisfies this inequality if and only if the order of 2 mod p^3 is the order of 2 mod p, which is equivalent to p^3 dividing 2^(p-1)-1, but no such prime p are known (as opposed to the A001220 Wieferich primes).

%H Charles R Greathouse IV, <a href="/A237663/b237663.txt">Table of n, a(n) for n = 1..10000</a>

%F Odd numbers m such that A002326((m^3-1)/2) < m * A002326((m^2-1)/2).

%F Odd numbers m such that 1 < gcd(A165781((m-1)/2), m) is a square.

%t okQ[m_] := MultiplicativeOrder[2, m^3] < m*MultiplicativeOrder[2, m^2]; Select[Range[1, 9999, 2], okQ] (* _Jean-François Alcover_, Dec 10 2015 *)

%o (PARI) is(m)=m%2 && znorder(Mod(2, m^3)) < m*znorder(Mod(2, m^2))

%Y Cf. A002326, A182297.

%K nonn

%O 1,1

%A _Thomas Ordowski_ and _Charles R Greathouse IV_, Feb 11 2014