On the decomposition of n| into prime powers
For a positive integer \(n\) consider a factorization of \(n!\) into a product of powers of \(n\) primes (not necessarily distinct). If \(p^\beta\) is the least power that occurs, represent \(\max\{p^\beta\}\) over all such factorizations as a power of \(n\), say \(n^{\alpha(n)}(0 <\alpha(n)<1)\). At the Western Number Theory Conference at UCLA in 1973 the question was raised as to the value of \(\varliminf\limits_n\alpha(n)\). The authors answer this question in a strong way. In fact, \(\lim\limits_n\alpha(n)=e^{\gamma-1}\) where \(\gamma=\sum\limits_{k=2}^\infty\frac 1k\log{\frac k{k-1}}\). This is a consequence of an asymptotic formula for the related function \[ \ell_n(\alpha)=\max\left\{k:n!=\prod\limits_{1\leq i\leq k}p_i^{\beta_i},p_i^{\beta_i}\geq n\right\} \] for \(0<\alpha<1\). Tne above lim inf is the largest \(\alpha\) for which \(\varliminf\ell_n(\alpha)/n\geq1\). It is proven that for a suitable constant \(b>0\) \[ \ell_n(\alpha)=(\gamma-\log\alpha)n+0(n/\exp(b\sqrt{\log\log n})). \]
scientific article; zbMATH DE number 3559680