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A038667 Minimal value of |product(A)-product(B)| where A and B are a partition of {1,2,3,...n}. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..35.

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) = 79200 - 78624 = 576.

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 (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])); 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<p, k++, If[j==2 || k==2 || k==1 && ll[[j-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-Fran├žois Alcover, Feb 29 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A061057, A200743, A200744, A263292.

Sequence in context: A036655 A098792 A061057 * A199823 A013608 A196441

Adjacent sequences:  A038664 A038665 A038666 * A038668 A038669 A038670

KEYWORD

nonn

AUTHOR

Erich Friedman

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

STATUS

approved

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Last modified March 27 08:41 EDT 2017. Contains 284146 sequences.