1) Which positive integers have exactly four positive divisors? 2) Let n be a positive integer. Show that $\displaystyle \tau(2^n-1) \geq \tau(n)$
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Originally Posted by Janu42 1) Which positive integers have exactly four positive divisors? 2) Let n be a positive integer. Show that $\displaystyle \tau(2^n-1) \geq \tau(n)$ 1) Integers of the form $\displaystyle \,p_1p_2$ or $\displaystyle \,p^3$. Because $\displaystyle \tau(n) = \prod(\alpha_i+1)$ for $\displaystyle n=p_1^{\alpha_1}\cdots$. 2) Haven't solved, except that it's obviously true for n prime.
For 2, note that if $\displaystyle d\mid n$ then $\displaystyle 2^d-1 \mid 2^n-1$
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