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A085817
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Start at (2n+1)/4 and iterate the map x -> x*ceiling(x); sequence gives values of n for which the denominators in the orbit drop directly from 4 to 1, bypassing 2.
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3
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4, 6, 7, 9, 12, 13, 14, 15, 20, 21, 22, 23, 24, 30, 31, 36, 38, 39, 41, 44, 45, 46, 47, 52, 53, 54, 55, 56, 62, 63, 69, 70, 71, 73, 76, 77, 78, 79, 81, 86, 87, 88, 94, 95, 96, 97, 101, 102, 103, 105, 108, 109, 110, 111, 112, 118, 119, 120, 126, 127, 129, 132, 133, 134, 135, 137
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