A200744 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the larger product.

1, 2, 3, 6, 12, 30, 72, 210, 630, 1920, 6336, 22176, 79200, 295680, 1146600, 4586400, 18869760, 80061696, 348986880, 1560176640, 7148445696, 33530112000, 160825785120, 787718131200, 3938590656000, 20083261440000, 104351247000000, 552173794099200, 2973528918360000, 16286983961149440

Offset: 1

Author

Franklin T. Adams-Watters, Nov 21 2011

Keywords

• nonn

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.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..140 (terms for n = 1..35 from Michael S. Branicky)

Crossrefs

Cf. A060777, A060796, A038667, A127180, A200743.

Programs

Maple

a:= proc(n) local l, ll, g, 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])); ll[n]/ g(1, nops(l), 1) end: seq(a(n), n=1..23);  # Alois P. Heinz, Nov 22 2011

Mathematica

a[n_] := a[n] = Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]}& /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]]&][[1]]; Max[Times @@ s, Times @@ t]];
Table[Print[n, " ", a[n]];
a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 07 2020 *)

Python

from math import prod, factorial
from itertools import combinations
def A200744(n):
    m = factorial(n)
    return min((abs((p:=prod(d))-m//p),max(p,m//p)) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l))[1] # Chai Wah Wu, Apr 07 2022

Formula

a(n) = A127180(n) - A200743(n) = A038667(n) + A200743(n) = (A038667(n) + A127180(n)) / 2. - Max Alekseyev, Jun 18 2022

Extensions

a(24)-a(30) from Alois P. Heinz, Nov 22 2011