An inequality between prime powers dividing n!

An Inequality was suggested by Balacenoiu at the First International Conference on Smarandache Notions in Number Theory

THE AVERAGE SMARANDACHE FUNCTION

Presenting an application to a diophantine equation. The application is not of the theorem per se, but rather of the counting method used to prove the theorem

ON THE DIVERGENCE OF THE SMARANDACHE HARMONIC SERIES

For any positive integer n let S(n) be the minimal positive integer m such that n I m!

Perfect Powers in Smarandache Type Expressions

The author showed that there are only finitely many numbers of the above form which are products of factorials

The rrth moment of the divisor function: an elementary approach

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$\sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}),$$ for any integer $r\ge 2$. Here, $$C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right).$$Comment: 8 pages, revise

Number Fields in Fibers: the Geometrically Abelian Case with Rational Critical Values

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.Comment: Some typos are corrected. The article is now accepted in Periodica Math. Hungaric