# [seqfan] Re: Partitioning 1..n

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Thu Feb 25 19:24:57 CET 2010

```A little off-topic, but the original question indeed reminds me of a "game" of mine.

See sequence A125584.
http://oeis.org/classic/A125584

There the two products are added, not subtracted. The goal of the game is to maximize the number of divisors of the sum.

Thanks,
Leroy Quet

[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]

--- On Thu, 2/25/10, hv at crypt.org <hv at crypt.org> wrote:

> From: hv at crypt.org <hv at crypt.org>
> Subject: Partitioning 1..n
> To: seqfan at list.seqfan.eu
> Cc: hv at crypt.org, q1qq2qqq3qqqq at yahoo.com
> Date: Thursday, February 25, 2010, 2:18 PM
> 1. Let a(n) = min(|prod(A) -
> prod(B)|) such that A, B partition { 1..n }.
> I calculate the sequence goes like:
>  1..10: 0 1 1 2 2 6 2 18 54 30
> 11..20: 36 576 576 840 6120 24480 20160 93696 420480
> 800640
> 21..30: 1305696 7983360 80313120 65318400 326592000
> 2286926400 3002360256 13680979200 37744574400 797369149440
> 31..  : 1763653953600 16753029012720 16880461678080
>
> Can someone confirm these values?
>
> Can we constrain upper or lower bounds for a(n), or say
>
> 2. Let a(n) = min(|lcm(A) - lcm(B)|) such that A, B
> partition { 1..n }.
> I calculate the sequence goes like:
>  1..10: 0 1 1 1 2 2 2 2 2 2
> 11..20: 25 25 60 60 60 35 24 24 372 372
> 21..  : 372 372 2802 2802 724 724
>
> Can someone confirm these values?
>
> Can we constrain upper bounds for a(n), or say anything
>
> Calculation: for #2, I fix 1 and 2, and try distributing
> 3..n in all the
> 2^(n-2) possible ways - I'm sure there must be a better
> way. For #1, I take
> the factors of n!, then from the centre of the list
> outwards search for
> a factor-pair that can be the products of an appropriate A
> and B.
>
> Neither of these sequences is in OEIS. I'll submit them in
> a few days, along
> with any additional material that arises in the interim.
>
> With apologies to Leroy Quet, there are a number of related
> sequences that can
> be expressed as games where players A and B take turns to
> select a number
> from the list, where A tries to either maximize or minimize
> one of f(A) - f(B)
> or |f(A) - f(B)|, with f() being prod or lcm, and in each
> case such that B
> aims to foil A. That would give 8 additional sequences with
> a(n) representing
> the best score A can achieve against best play. I haven't
> attempted to
> calculate those sequences, but may do so if I get the time
> - I welcome anyone
> else that wants to calculate and submit them.
>
> Hugo
>

```