Erdős Problem 304: are all fractions $a/b$ sums of $O(\log\log b)$ distinct unit fractions?

Theorem 1.1: there are absolute $c_1,c_2>0$ and $b_0$ with $c_1\log\log b\le N(b)\le c_2\log\log b$ for $b\ge b_0$; the upper bound holds for every numerator. Corollaries: the number $F(k)$ of $k$-term expansions of 1 satisfies $\log\log F(k)\asymp k$ (Erdős-Graham asked for estimates), and the least integer $v(k)\ge2$ missing from all $k$-term expansions of 1 satisfies $e^{e^{k/600}}\le v(k)\le 1+k^{2^{k-1}}$ eventually, so $\log\log v(k)\asymp k$, answering the growth question of Erdős Problem 293 up to constants. Not shown: the optimal constants, or control of denominator sizes (unlike Tenenbaum-Yokota).

Theorem 1.1: there are absolute $c_1,c_2>0$ and $b_0$ with $c_1\log\log b\le N(b)\le c_2\log\log b$ for $b\ge b_0$; the upper bound holds for every numerator. Corollaries: the number $F(k)$ of $k$-term expansions of 1 satisfies $\log\log F(k)\asymp k$ (Erdős-Graham asked for estimates), and the least integer $v(k)\ge2$ missing from all $k$-term expansions of 1 satisfies $e^{e^{k/600}}\le v(k)\le 1+k^{2^{k-1}}$ eventually, so $\log\log v(k)\asymp k$, answering the growth question of Erdős Problem 293 up to constants. Not shown: the optimal constants, or control of denominator sizes (unlike Tenenbaum-Yokota).