# [seqfan] Partitioning 1..n

hv at crypt.org hv at crypt.org
Thu Feb 25 15:18:50 CET 2010

```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 anything else about it?

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 else about it?

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

```