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A127180
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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)}.
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6
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2, 2, 3, 5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000, 7876854720000, 40164235953600
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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