

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
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OFFSET

0,1


COMMENTS

Answering a question asked by Leroy Quet in rec.puzzles on 20070105.
The terms were also calculated by Peter Pein and Jan Kristian Haugland.


LINKS

Table of n, a(n) for n=0..25.
Leroy Quet, MultiplyThenAdd "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 12divisor 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}] (* JeanFrançois 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, n2), 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



