A282552 Difference between the multiplicative orders of 2 modulo p^2 and 2 modulo p, where p = prime(n).

4, 16, 18, 100, 144, 128, 324, 242, 784, 150, 1296, 800, 588, 1058, 2704, 3364, 3600, 4356, 2450, 648, 3042, 6724, 968, 4608, 10000, 5202, 11236, 3888, 3136, 882, 16900, 9248, 19044, 21904, 2250, 8112, 26244, 13778, 29584, 31684, 32400, 18050, 18432, 38416

Offset: 2

Author

Felix Fröhlich, Feb 18 2017

Keywords

• nonn

Comments

a(n) = 0 iff A014664(n) = A243905(n), i.e., iff prime(n) is a Wieferich prime (A001220). So far this is known to be the case only for prime(183) = 1093 and prime(490) = 3511, i.e., a(183) = 0 and a(490) = 0.

Crossrefs

Cf. A001220, A014664, A243905.

Programs

Mathematica

Table[MultiplicativeOrder[2, #^2] - MultiplicativeOrder[2, #] &@ Prime@ n, {n, 2, 45}] (* Michael De Vlieger, Feb 18 2017 *)

PARI

a(n) = my(p=prime(n)); znorder(Mod(2, p^2)) - znorder(Mod(2, p))

Formula

a(n) = A243905(n) - A014664(n).