A237663 Odd numbers m such that the order of 2 mod m^3 is less than m times the order of 2 mod m^2.

57, 111, 219, 285, 327, 399, 489, 505, 543, 555, 597, 627, 741, 777, 813, 969, 1083, 1095, 1137, 1221, 1255, 1299, 1311, 1379, 1425, 1443, 1461, 1467, 1515, 1533, 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

Offset: 1

Author

Thomas Ordowski and Charles R Greathouse IV, Feb 11 2014

Keywords

• nonn

Comments

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

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).

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002326, A182297.

Programs

Mathematica

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

PARI

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

Formula

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

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