A200744 - OEIS

%I #43 Jul 15 2023 16:57:14

%S 1,2,3,6,12,30,72,210,630,1920,6336,22176,79200,295680,1146600,

%T 4586400,18869760,80061696,348986880,1560176640,7148445696,

%U 33530112000,160825785120,787718131200,3938590656000,20083261440000,104351247000000,552173794099200,2973528918360000,16286983961149440

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

%H Max Alekseyev, <a href="/A200744/b200744.txt">Table of n, a(n) for n = 1..140</a> (terms for n = 1..35 from Michael S. Branicky)

%F a(n) = A127180(n) - A200743(n) = A038667(n) + A200743(n) = (A038667(n) + A127180(n)) / 2. - _Max Alekseyev_, Jun 18 2022

%e 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.

%e 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.

%p 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])); ll[n]/ g(1, nops(l), 1) end: seq(a(n), n=1..23);  # _Alois P. Heinz_, Nov 22 2011

%t 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]];

%t Table[Print[n, " ", a[n]];

%t a[n], {n, 1, 25}] (* _Jean-François Alcover_, Nov 07 2020 *)

%o (Python)

%o from math import prod, factorial

%o from itertools import combinations

%o def A200744(n):

%o     m = factorial(n)

%o     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

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

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Nov 21 2011

%E a(24)-a(30) from _Alois P. Heinz_, Nov 22 2011