This question got no attention these past few days

Number Theory Level 3

Define $HF(n)$ as the hyperfactorial function of $n$ , where $\displaystyle HF(n) = \prod_{m=1}^n m^m$ .

For what integer $n$ , is $HF(n)$ equal to the number of milliseconds in a day?

Try to solve this without using a calculator.

Inspiration .

The answer is 5.

1 solution

Chew-Seong Cheong
Jul 7, 2015

The number of milliseconds in a day

$\begin{aligned} M & = 24\times 60 \times 60 \times 1000 \\ & = 2^3\dot{} 3 \times 2^2\dot{} 3 \dot{} 5 \times 2^2\dot{} 3 \dot{} 5 \times 2^3\dot{} 5^3 \\ & = 2^{10} \dot{} 3^3 \dot{} 5^5 \\ & = 1^1 \dot{} 2^2\dot{} 3^3 \dot{} 4^4 \dot{} 5^5 \\ & = \prod_{m=1}^5 m^m = HF (\boxed{5}) \end{aligned}$

Moderator note:

Simple standard approach.

Nice solution. @Pi Han Goh Fun result. Just thought I should mention that the function should read

$HF(n) = \displaystyle\prod_{m=1}^{n} m^{m}.$

Brian Charlesworth - 5 years, 11 months ago

Okie fixed thanks

Pi Han Goh - 5 years, 11 months ago

This question got no attention these past few days

The answer is 5.

1 solution

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