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
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