Claims

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1944_07_01_erdos: Erdős's 1944 theorem that consecutive highly composite numbers lie within a factor $1+(\log n)^{-c}$ of each other, so more than $(\log x)^{1+c}$ of them lie below $x$; it answers the question yes for the exponents $k\le1+c$.

1971_02_01_nicolas: Nicolas's 1971 upper bound Q(X) = O((log X)^(1+c)) on the number of highly composite numbers up to X, which answers the question no: the count stays below a fixed power of the logarithm.

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