A219209 Maximal product of all parts of a partition of n into distinct divisors of n.

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 48, 13, 14, 15, 16, 17, 162, 19, 200, 21, 22, 23, 1152, 25, 26, 27, 784, 29, 1350, 31, 32, 33, 34, 35, 15552, 37, 38, 39, 6400, 41, 2058, 43, 44, 45, 46, 47, 73728, 49, 50, 51, 52, 53, 8748, 55, 25088, 57, 58, 59, 864000

Offset: 0

Author

Alois P. Heinz, Nov 14 2012

Keywords

• nonn

Examples

			a(0) = 1: the empty product.
a(p) = p for any prime p: [p]-> p.
a(12) = 48: [2,4,6]-> 48.
a(20) = 200: [1,4,5,10]-> 200.
a(24) = 1152: [1,2,3,4,6,8]-> 1152.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

The number of distinct products are in A219208.

Cf. A033630, A092976.

Programs

Maple

a:= proc(n) local b, l;
      l:= sort([numtheory[divisors](n)[]]);
      b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
            max(b(n, i-1), `if`(l[i]>n, 0, l[i] *b(n-l[i], i-1)))))
          end; forget(b);
      b(n, nops(l))
    end:
seq(a(n), n=0..80);

Mathematica

a[n_] := a[n] = Module[{b, l}, l = Divisors[n]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i<1, 0, Max[b[m, i-1], If[l[[i]]>m, 0, l[[i]]*b[m-l[[i]], i-1] ]]]]; b[n, Length[l]]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)