A single formula to express every positive integer with digits 0-9 in which every digit is used only once.

Ross Honsberger (Mathematical Gems III, MAA, 1985, p 175) credits Verner Hoggatt Jr. with the following formula

,

where there are exactly n square roots. If the fraction in the first base is written as (0 + 1 + 2 + 3 + 4)/5, then all digits enter the formula in their natural order. Simplified the formula appears as

,

which is equivalent to

,

which, in turn, states the obvious


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