A125097 a(n) is k such that k*(2n+1) mod 1000 = 123, or 2n+1 if no such k exists.

123, 41, 5, 589, 347, 193, 471, 15, 419, 217, 863, 701, 25, 449, 487, 133, 731, 35, 679, 157, 3, 561, 45, 109, 227, 473, 191, 55, 739, 697, 543, 621, 65, 569, 567, 213, 851, 75, 599, 837, 483, 881, 85, 829, 507, 353, 711, 95, 259, 577, 823, 341, 105, 889, 47, 893