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Curious Inequality 
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#1
Feb1904, 10:38 AM

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


#2
Feb1904, 10:52 AM

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