Curious Inequality

by MathNerd
Tags: curious, inequality
 P: n/a I know that this isn’t very practical but I discovered the following curious inequality when I was playing around with $$d(n)$$ where $$d(n)$$ gives the number of divisors of $$n \ \epsilon \ N$$. If $$n$$ has $$p$$ prime factors (doesn’t have to be distinct prime factors e.g. $$12 = 2^2 \ 3$$ has got three prime factors (2,2,3)), Then $$p + 1 \leq d(n) \leq \sum_{k=0}^{p} _{p} C_{k}$$ I don’t know if this has been previously discovered but giving its simplicity it wouldn’t surprise me if it has.
 Originally posted by MathNerd I know that this isn’t very practical but I discovered the following curious inequality when I was playing around with $$d(n)$$ where $$d(n)$$ gives the number of divisors of $$n \ \epsilon \ N$$. If $$n$$ has $$p$$ prime factors (doesn’t have to be distinct prime factors e.g. $$12 = 2^2 \ 3$$ has got three prime factors (2,2,3)), Then $$d(n) \leq \sum_{k=0}^{p} _{p} C_{k}$$ I don’t know if this has been previously discovered but giving its simplicity it wouldn’t surprise me if it has.