OFFSET
0,1
COMMENTS
The maximum (product of b(k)'s + product of c(k)'s) occurs, for n>=2, when {b(k)} = (2,3,4,...n) and {c(k)} = (1). a(1) = 2 because the product over the empty set is defined here as 1.
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..140
FORMULA
a(n) <= A060696(n+1) = A076051(n) considering the interleaved partition b={2,4,6,..}, c={1,3,5, 7,...}. - R. J. Mathar, Jan 10 2007
EXAMPLE
By partitioning (1,2,3,...8) into {b(k)} and {c(k)} so that {b(k)} = (1,4,6,8) and {c(k)} = (2,3,5,7), then (product of b(k)'s + product of c(k)'s) is minimized. Therefore a(8) = 1*4*6*8 + 2*3*5*7 = 402.
MAPLE
LQprod := proc(S) if nops(S) = 0 then 1 ; else product(S[i], i=1..nops(S)) ; fi ; end: A127180 := proc(n) local S, m, B, b, c, s, res, i ; res := -1 ; S := {} ; for i from 1 to n do S := S union {i} ; od; for m from 0 to n/2 do B := combinat[permute](n, m) ; for i from 1 to nops(B) do b := op(i, B) ; c := S minus convert(b, set) ; s := LQprod(b)+LQprod(c) ; if res < 0 or s < res then res := s ; fi ; od ; od ; RETURN(res) ; end: for n from 1 to 20 do A127180(n) ; od ; # R. J. Mathar, Jan 10 2007
MATHEMATICA
a[n_] := a[n] = Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]}& /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]]&][[1]]; Times @@ s + Times @@ t];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 24}] (* Jean-François Alcover, May 06 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Jan 07 2007
EXTENSIONS
a(9)-a(13) from R. J. Mathar, Jan 10 2007
a(14)-a(26) from Ray Chandler, Feb 14 2007
STATUS
approved