%I A125584
%S A125584 2,2,2,2,4,4,12,20,16,24,64,96,144,128,320,384,512,1008,1296,1024,2700,
%T A125584 2592,4800
%N A125584 Maximum number of divisors of Prod(a_i) + Prod(b_j) over all (disjoint) 
               partitions of {1..n} into {a_i} and {b_j}.
%C A125584 Answering a question asked by Leroy Quet in rec.puzzles on 2007-01-05.
%C A125584 The terms were also calculated by Peter Pein and J. K. Haugland.
%H A125584 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> 
               (listed in lieu of email address)
%H A125584 L. Quet, <a href="http://groups.google.com.au/group/rec.puzzles/msg/261178ba23034b9b">
               Multiply-Then-Add "Game"</a>, USENET post to rec.puzzles.
%e A125584 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).
%e A125584 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.
%e A125584 a(7) = 20 because of the partition 3*4 + 2*5*6*7 = 432, which has 20 
               divisors (and no other partition yields more).
%p A125584 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 (mathar(AT)strw.leidenuniv.nl), 
               Nov 11 2007
%o A125584 (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] ];
%Y A125584 Sequence in context: A010334 A010578 A005866 this_sequence A029078 A131799 
               A078635
%Y A125584 Adjacent sequences: A125581 A125582 A125583 this_sequence A125585 A125586 
               A125587
%K A125584 nonn
%O A125584 0,1
%A A125584 Geoff Bailey (geoff(AT)maths.usyd.edu.au), Jan 04 2007
%E A125584 2 more terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 
               2007
%E A125584 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 03 2008 at 
               the suggestion of R. J. Mathar