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

%I A219209 #15 Feb 16 2017 11:00:50
%S A219209 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,
%T A219209 25,26,27,784,29,1350,31,32,33,34,35,15552,37,38,39,6400,41,2058,43,
%U A219209 44,45,46,47,73728,49,50,51,52,53,8748,55,25088,57,58,59,864000
%N A219209 Maximal product of all parts of a partition of n into distinct divisors of n.
%H A219209 Alois P. Heinz, <a href="/A219209/b219209.txt">Table of n, a(n) for n = 0..10000</a>
%e A219209 a(0) = 1: the empty product.
%e A219209 a(p) = p for any prime p: [p]-> p.
%e A219209 a(12) = 48: [2,4,6]-> 48.
%e A219209 a(20) = 200: [1,4,5,10]-> 200.
%e A219209 a(24) = 1152: [1,2,3,4,6,8]-> 1152.
%p A219209 a:= proc(n) local b, l;
%p A219209       l:= sort([numtheory[divisors](n)[]]);
%p A219209       b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A219209             max(b(n, i-1), `if`(l[i]>n, 0, l[i] *b(n-l[i], i-1)))))
%p A219209           end; forget(b);
%p A219209       b(n, nops(l))
%p A219209     end:
%p A219209 seq(a(n), n=0..80);
%t A219209 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 *)
%Y A219209 The number of distinct products are in A219208.
%Y A219209 Cf. A033630, A092976.
%K A219209 nonn
%O A219209 0,3
%A A219209 _Alois P. Heinz_, Nov 14 2012