cp's OEIS Frontend

A266871 Number of partitions of n that maximize the product of multiplicities of parts.

%I A266871 #18 Dec 21 2016 03:16:31
%S A266871 1,1,1,1,1,1,1,1,2,1,2,1,1,1,2,1,1,1,2,1,1,1,1,3,1,1,1,2,1,1,1,3,1,1,
%T A266871 2,1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,2,1,1,3,1,1,1,5,1,1,1,2,1,1,1,2,1,1,
%U A266871 3,1,1,1,5,1,1,1,2,1,1,1,3,1,1,2,1,1,1,1,2,1,1,1,1,1,2,1,1,1,3,1,1,2,1,1,1,1,2,1,1,1,4,1,1,1,2,1
%N A266871 Number of partitions of n that maximize the product of multiplicities of parts.
%H A266871 Alois P. Heinz, <a href="/A266871/b266871.txt">Table of n, a(n) for n = 0..5000</a>
%F A266871 a(n) = A266477(n,A266480(n)).
%e A266871 a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2] (product of multiplicities = 8).
%e A266871 a(9) = 1: [1,1,1,1,1,2,2] (product = 10).
%e A266871 a(10) = 2: [1,1,1,1,1,1,2,2], [1,1,1,1,2,2,2] (product = 12).
%e A266871 a(11) = 1: [1,1,1,1,1,2,2,2] (product = 15).
%e A266871 a(23) = 3: [1,1,1,1,1,1,1,1,1,2,2,2,2,3,3], [1,1,1,1,1,1,1,1,2,2,2,3,3,3], [1,1,1,1,1,1,2,2,2,2,3,3,3] (product = 72).
%p A266871 b:= proc(n, i) option remember; local r,l,j;
%p A266871       if n=0 or i=1 then [max(1, n),1]
%p A266871     else r:= b(n, i-1);
%p A266871          for j to iquo(n, i) do
%p A266871            l:= (w-> [w[1]*j, w[2]])(b(n-i*j, i-1));
%p A266871            r:= `if`(l[1]>r[1], l,
%p A266871                `if`(l[1]=r[1], [0, l[2]], 0)+r)
%p A266871          od; r
%p A266871       fi
%p A266871     end:
%p A266871 a:= n-> b(n$2)[2]:
%p A266871 seq(a(n), n=1..120);
%t A266871 b[n_, i_] := b[n, i] = Module[{r, l, j}, If[n == 0 || i == 1, {Max[1, n], 1}, r = b[n, i - 1]; For[j = 1, j <= Quotient[n, i], j++, l = Function[w, {w[[1]]*j, w[[2]]}][b[n - i*j, i - 1]]; r = If[l[[1]] > r[[1]], l, If[l[[1]] == r[[1]], {0, l[[2]]}, 0] + r]]; r]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 120}] (* _Jean-François Alcover_, Dec 21 2016, translated from Maple *)
%Y A266871 Cf. A266477, A266480.
%K A266871 nonn
%O A266871 0,9
%A A266871 _Emeric Deutsch_ and _Alois P. Heinz_, Jan 04 2016