OFFSET
1,3
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..140 (terms for n=1..35 from Michael S. Branicky)
FORMULA
EXAMPLE
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) = 78624.
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 (mm<p) do if j=2 or k=2 or k=1 and ll[j-1]*mm>bb 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])); 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]]; Min[Times @@ s, Times @@ t]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 03 2020 *)
PROG
(Python)
from itertools import combinations
def prod(l):
t=1
for x in l:
t *= x
return t
def a200743(n):
nums = list(range(1, n+1))
widths = combinations(nums, n//2)
dimensions = [(prod(width), prod(x for x in nums if x not in width)) for width in widths]
best = min(dimensions, key=lambda x:max(*x)-min(*x))
return min(best)
# Christian Perfect, Feb 04 2015
(Python)
from math import prod, factorial
from itertools import combinations
def A200743(n):
m = factorial(n)
return min((abs((p:=prod(d))-m//p), min(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Nov 21 2011
EXTENSIONS
a(24)-a(30) from Alois P. Heinz, Nov 22 2011
a(31) from Michael S. Branicky, May 21 2021
STATUS
approved