|
%I #19 May 08 2023 09:34:26
%S 2,2,3,5,10,22,54,142,402,1206,3810,12636,43776,157824,590520,2287080,
%T 9148320,37719360,160029696,697553280,3119552640,14295585696,
%U 67052240640,321571257120,1575370944000,7876854720000,40164235953600
%N a(n) = smallest possible (product of b(k)'s + product of c(k)'s), where the positive integers <= n are partitioned somehow into {b(k)} and {c(k)}.
%C 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.
%H Max Alekseyev, <a href="/A127180/b127180.txt">Table of n, a(n) for n = 0..140</a>
%F 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
%F a(n) = A200743(n) + A200744(n) = (A200744(n)^2 - A200743(n)^2) / A038667(n). - _Max Alekseyev_, Apr 08 2022
%e 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.
%p 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
%t 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];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 24}] (* _Jean-François Alcover_, May 06 2023 *)
%Y Cf. A038667, A127181, A200743, A200744,
%K nonn
%O 0,1
%A _Leroy Quet_, Jan 07 2007
%E a(9)-a(13) from _R. J. Mathar_, Jan 10 2007
%E a(14)-a(26) from _Ray Chandler_, Feb 14 2007
|
