partition that maximizes lcm

Asked Aug 28 '19 at 11:50

Active Aug 28 '19 at 11:50

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Let $n$ be a natural number. How can I find a partition of $n$ such that its least common multiple is maximized?

In other words, how can I find $a_1, a_2, \cdots a_m \in \mathbb{N}$ such that $$a_1 + a_2 + \cdots + a_m = n \\\text{and} \\ \text{lcm}(a_1, a_2, \cdots ,a_m) \text{ is largest}$$ ?

If not possible to find it, is the largest lcm value bounded by some functions of n? I tried to find it's upper bound by multiplying all numbers from each partition, and using a little calculus, able to drive the upper bound $e^{n/e}$. But it seems a bit too loose. Is there a better bound for lcm?

asked Aug 28 '19 at 11:50

coxehj4142

are all parts to be unique ? – Aug 28 '19 at 12:22
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This seems to be equivalent to finding the maximum order of any element in the symmetric group $S_n$. So this may be helpful. https://math.stackexchange.com/questions/221211/maximal-order-of-an-element-in-a-symmetric-group – paw88789 Aug 28 '19 at 12:22
Wow, thanks @paw88789. Your answer was what I was looking for. – coxehj4142 Aug 29 '19 at 11:45
@Roddy MacPhee what do you mean by unique?? – coxehj4142 Aug 29 '19 at 11:46
distinct not taking the same value. – Aug 29 '19 at 11:48
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Possible duplicate of Maximal order of an element in a symmetric group – Peter Taylor Aug 30 '19 at 08:21
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@Roddy MacPhee No need to, but since we're trying to maximize lcm, I think each number is likely to be distinct in the end. – coxehj4142 Aug 30 '19 at 15:27
@Peter Taylor Thanks, that is what I wanted! But also, do you have any idea for calculating the exact partition that lcm is maximized? – coxehj4142 Aug 30 '19 at 15:31
I have a definite memory of writing a program to calculate these numbers for some challenge site - probably Project Euler - but I can't remember enough to find the program on my hard disk. Nevertheless, I think it's probably easy to prove that the partition should be into prime powers (plus maybe a small useless remnant), and then a recursive search would be reasonably efficient. – Peter Taylor Aug 30 '19 at 15:52
@Peter Taylor Thanks for your help!! It was very helpful. I'll try proving the prime power property. – coxehj4142 Aug 31 '19 at 07:04

partition that maximizes lcm

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