New complete Latin squares of odd order

Let m be an odd powerful number. We show that there is a sequenceable group of order 3m and hence a complete Latin square of that order. Further, the sequencings we construct are starter-translate and so they may be combined with themselves and other sequencings to construct complete Latin squares for many more orders. We also consider square-free m with all prime divisors congruent to 1 modulo 6: there is a sequenceable group, and hence a complete Latin square, of order 3m in this case too. This work gives 104 new orders less than 10,000 for which a complete Latin square is known to exist, the smallest of which is 75.