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Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?

(This question arose during a discussion with participants in the Soacha (Colombia) 2023 Math Circle.)

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?

(This question arose during a discussion with participants in the Soacha (Colombia) 2023 Math Circle.)

499 and the Gamma Function

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?

(This question arose during a discussion with participants in the Soacha (Colombia) 2023 Math Circle.)

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials, fractional points (e.g. 0.0001111), and square roots?

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?