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 A125584 Maximum number of divisors of Product(a_i) + Product(b_j) over all (disjoint) partitions of {1..n} into {a_i} and {b_j}. 1
 2, 2, 2, 2, 4, 4, 12, 20, 16, 24, 64, 96, 144, 128, 320, 384, 512, 1008, 1296, 1024, 2700, 2592, 4800, 6144, 8448, 12672 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Answering a question asked by Leroy Quet in rec.puzzles on 2007-01-05. The terms were also calculated by Peter Pein and Jan Kristian Haugland. LINKS Leroy Quet, Multiply-Then-Add "Game", USENET post to rec.puzzles. EXAMPLE a(1) = 2 because the product over the empty set is defined here as 1. So we have a(1) = number of divisors of (1+1). For n = 6 the maximum number of divisors occurs when S = 1*3*4*5 + 2*6 = 72. (This 12-divisor solution is not unique.) So a(6) is the number of positive divisors of 72, which is 12. a(7) = 20 because of the partition 3*4 + 2*5*6*7 = 432, which has 20 divisors (and no other partition yields more). MAPLE A125584 := proc(n) local bc, a, b, c, i, j, bL, S, bsiz ; a := 0 ; bc := {seq(i, i=1..n)} ; for bsiz from 0 to floor(n/2) do bL := combinat[choose](bc, bsiz) ; for i from 1 to nops(bL) do b := convert(op(i, bL), set) ; c := bc minus b ; if nops(b) = 0 then b := 1; else b := mul(j, j=b) ; fi ; if nops(c) = 0 then c := 1; else c := mul(j, j=c) ; fi ; S := numtheory[tau](c+b) ; a := max(a, S) ; od: od: RETURN(a) ; end: for n from 1 do A125584(n) ; od; # R. J. Mathar, Nov 11 2007 MATHEMATICA a[n_] := a[n] = Table[DivisorSigma[0, Times @@ sr + Times @@ Complement[ Range[n], sr]], {sr, Subsets[Range[n], n]}] // Max; Table[Print[n, " ", a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 10 2020 *) PROG (MAGMA) [ n lt 3 select 2 else Max([NumberOfDivisors(x + (Factorial(n) div x)) where x is &*s : s in Subsets({3..n}) ] : n in [0..20] ]; (PARI) a(n)={my(m=0); forsubset(max(0, n-2), s, my(t=prod(i=1, #s, s[i]+1)); m=max(m, numdiv(t + n!/t))); m} \\ Andrew Howroyd, Jan 28 2020 CROSSREFS Sequence in context: A328583 A240674 A005866 * A230447 A029078 A131799 Adjacent sequences:  A125581 A125582 A125583 * A125585 A125586 A125587 KEYWORD nonn,more AUTHOR Geoff Bailey (geoff(AT)maths.usyd.edu.au), Jan 04 2007 EXTENSIONS 2 more terms from R. J. Mathar, Nov 11 2007 Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar a(23)-a(25) from Andrew Howroyd, Jan 28 2020 STATUS approved

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)