Positive Integers Expressible by Sum of Integers whose Reciprocals Sum to 1

Theorem
Every positive integer over $77$ can be expressed as the sum of positive integers whose reciprocals add up to $1$.

The full sequence of numbers that cannot be expressed as such is:
 * $2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77$