Curious Inequality

[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

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.

[matt grime]

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.

the sum you wrote down is just 2^p btw. and isn't that result rather obvious? I mean p distinct primes gives you 2^p divisors, so repeated primes naturally gives you fewer.