cp's OEIS Frontend

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

%I A200744 #43 Jul 15 2023 16:57:14
%S A200744 1,2,3,6,12,30,72,210,630,1920,6336,22176,79200,295680,1146600,
%T A200744 4586400,18869760,80061696,348986880,1560176640,7148445696,
%U A200744 33530112000,160825785120,787718131200,3938590656000,20083261440000,104351247000000,552173794099200,2973528918360000,16286983961149440
%N A200744 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the larger product.
%H A200744 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 A200744 a(n) = A127180(n) - A200743(n) = A038667(n) + A200743(n) = (A038667(n) + A127180(n)) / 2. - _Max Alekseyev_, Jun 18 2022
%e A200744 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 A200744 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 A200744 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 A200744 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 A200744 Table[Print[n, " ", a[n]];
%t A200744 a[n], {n, 1, 25}] (* _Jean-François Alcover_, Nov 07 2020 *)
%o A200744 (Python)
%o A200744 from math import prod, factorial
%o A200744 from itertools import combinations
%o A200744 def A200744(n):
%o A200744     m = factorial(n)
%o A200744     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 A200744 Cf. A060777, A060796, A038667, A127180, A200743.
%K A200744 nonn
%O A200744 1,2
%A A200744 _Franklin T. Adams-Watters_, Nov 21 2011
%E A200744 a(24)-a(30) from _Alois P. Heinz_, Nov 22 2011