A038667 Minimal value of |product(A) - product(B)| where A and B are a partition of {1,2,3,...,n}.

0, 0, 1, 1, 2, 2, 6, 2, 18, 54, 30, 36, 576, 576, 840, 6120, 24480, 20160, 93696, 420480, 800640, 1305696, 7983360, 80313120, 65318400, 326592000, 2286926400, 3002360256, 13680979200, 37744574400, 797369149440, 1763653953600, 16753029012720, 16880461678080, 10176199188480, 26657309952000

Offset: 0

Author

Erich Friedman

Keywords

• nonn

Comments

Conjecture: The sequence of rational numbers A061057(n)/a(n) has 1 as a limit point. Question: What other limit points does the sequence have? - Richard Peterson, Jul 13 2023

Examples

			For n=1, we put 1 in one set and the other is empty; with the standard convention for empty products, both products are 1.
For n=13, the central pair of divisors of n! are 78975 and 78848. Since neither is divisible by 10, these values cannot be obtained. The next pair of divisors are 79200 = 12*11*10*6*5*2*1 and 78624 = 13*9*8*7*4*3, so a(13) = 79200 - 78624 = 576.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..140 (terms for n = 0..60 from Peter J. Taylor)

Crossrefs

Cf. A061033, A061057, A061060, A200743, A200744, A127180, A263292.

Programs

Maple

a:= proc(n) local l, ll, g, gg, p, i; l:= [i$i=1..n]; ll:= [i!$i=1..n]; g:= proc(m, j, b) local mm, bb, k; if j=1 then m else mm:= m; bb:= b; for k to 2 while (mmbb then bb:= max(bb, g(mm, j-1, bb)) fi; mm:= mm*l[j] od; bb fi end; Digits:= 700; p:= ceil(sqrt(ll[n])); gg:= g(1, nops(l), 1); ll[n]/gg -gg end: a(0):=0:
seq(a(n), n=0..20); #  Alois P. Heinz, Jul 09 2009, revised Oct 17 2015

Mathematica

a[n_] := Module[{l, ll, g, gg, p, i}, l = Range[n]; ll = Array[Factorial, n]; g[m_, j_, b_] := g[m, j, b] = Module[{mm, bb, k}, If[j==1, m, mm=m; bb=b; For[k=1, mm bb , bb = Max[bb, g[mm, j-1, bb]]]; mm = mm*l[[j]]]; bb]]; p = Ceiling[Sqrt[ ll[[n]]]]; gg = g[1, Length[l], 1]; ll[[n]]/gg - gg]; a[0]=0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 35}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

Python

from math import prod, factorial
from itertools import combinations
def A038667(n):
    m = factorial(n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l)) # Chai Wah Wu, Apr 06 2022

Formula

a(n) = A200744(n) - A200743(n) = (A200744(n)^2 - A200743(n)^2) / A127180(n). - Max Alekseyev, Apr 08 2022

a(n) >= A061057(n).

Extensions

a(28)-a(31) from Alois P. Heinz, Jul 09 2009

a(1) and examples from Franklin T. Adams-Watters, Nov 22 2011

a(32)-a(33) from Alois P. Heinz, Nov 23 2011

a(34)-a(35) from Alois P. Heinz, Oct 17 2015