A094312 Beginning with 1, re-arrangement of natural numbers such that {a(2n-1)*a(2n)}^(1/2) and {a(2n)+a(2n+1)}/2 are integers.

1, 4, 2, 8, 6, 216, 10, 1000, 12, 3, 5, 125, 7, 343, 9, 16, 14, 2744, 18, 32, 20, 45, 11, 1331, 13, 2197, 15, 3375, 17, 4913, 19, 6859, 21, 9261, 23, 12167, 25, 49, 27, 243, 29