0,1

Answering a question asked by Leroy Quet in rec.puzzles on 2007-01-05.

The terms were also calculated by Peter Pein and J. K. Haugland.

Leroy Quet, Home Page (listed in lieu of email address)

L. Quet, Multiply-Then-Add "Game", USENET post to rec.puzzles.

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).

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

(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] ];

Sequence in context: A010334 A010578 A005866 this_sequence A029078 A131799 A078635

Adjacent sequences: A125581 A125582 A125583 this_sequence A125585 A125586 A125587

nonn

Geoff Bailey (geoff(AT)maths.usyd.edu.au), Jan 04 2007

2 more terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 2007

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 03 2008 at the suggestion of R. J. Mathar