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Phi Number System

For every positive integer , there is a unique finite sequence of distinct nonconsecutive (not necessarily positive) integers , ..., such that(1)where is the golden ratio.For example, for the first few positive integers,(2)(3)(4)(5)(6)(7)(8)(OEIS A104605).The numbers of terms needed to represent for , 2, ... are given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, ... (OEIS A055778), which are also the numbers of 1s in the base- representation of .The following tables summarizes the values of that require exactly powers of in their representations.OEISnumbers requiring exactly powers2A0052482, 3, 7, 18, 47, 123, 322, 843, ...3A1046264, 5, 6, 8, 19, 48, 124, 323, 844, ...4A1046279, 10, 12, 13, 14, 16, 17, 20, 21, 25, ...5A10462811, 15, 22, 23, 24, 26, 30, 31, 32, 34, ...

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