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A348599 Irregular triangle T(n,k) where row n is the partition of n with the least number of 3-smooth parts such that the product of parts is maximal.

%I A348599 #21 Jun 20 2025 17:35:44
%S A348599 1,2,3,4,3,2,6,4,3,8,9,6,4,8,3,12,9,4,8,6,9,6,16,9,8,18,16,3,12,8,12,
%T A348599 9,16,6,9,8,6,24,16,9,18,8,27,16,12,27,2,18,12,27,4,32,24,9,18,16,27,
%U A348599 8,36,36,1,32,6,27,12,24,16,32,9,24,18,27,16,32,12,27,18
%N A348599 Irregular triangle T(n,k) where row n is the partition of n with the least number of 3-smooth parts such that the product of parts is maximal.
%C A348599 Let k = 2^a * 3^b be a part such that n is the sum of at least one such k. Then we may represent k at (a,b) on a Cartesian grid. In Dimitrov, et al., these k are known as "digits" in a 2-dimensional number base system. Since these partitions have the least number of parts (digits) in order to represent n, in Dimitrov this is called a "canonic form" for n in base (2,3).
%C A348599 These numbers represent an extreme representation where the digits tend to be placed close together.
%D A348599 Vassil Dimitrov, Graham Jullien, and Roberto Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press (2012), 35-39.
%H A348599 Michael De Vlieger, <a href="/A348599/b348599.txt">Table of n, a(n) for n = 1..10093</a> (rows n = 1..3600, flattened)
%H A348599 Michael De Vlieger, <a href="/A348599/a348599.png">Plot of parts in row n at (T(n,k), n)</a>, for n = 1..256.
%H A348599 Michael De Vlieger, <a href="/A348599/a348599_1.png">Plot T(n,k) at (T(n,k), n)</a> for n = 1..10000.
%H A348599 Michael De Vlieger, <a href="/A348599/a348599_2.png">Annotated plot of m = A347860(n,k) and m = T(n,k) at (m, n)</a> for n = 1..64, showing m in row n of this sequence in blue, m in row n of A347860 in red, but in black if these coincide.
%F A348599 A237442(n) = length of row n.
%e A348599 Triangle begins:
%e A348599    1;
%e A348599    2;
%e A348599    3;
%e A348599    4;
%e A348599    3, 2;    (product larger than (4,1))
%e A348599    6;
%e A348599    4, 3;    (product larger than (6,1))
%e A348599    8;
%e A348599    9;
%e A348599    6, 4;    (product greatest of {(9,1), (8,2), (6,4)})
%e A348599    8, 3;    (product larger than (9,2))
%e A348599   12;
%e A348599   ...
%t A348599 nn = 45; ss = Union@ Flatten@ Table[2^a*3^b, {a, 0, Log2[nn]}, {b, 0, Log[3, nn/(2^a)]}]; Table[Block[{k = 1, w, t = TakeWhile[ss, # <= n &]}, While[{} == Set[w, IntegerPartitions[n, {k}, t]], k++]; MaximalBy[w, Times @@ # &][[1]]], {n, nn}] // Flatten
%Y A348599 Cf. A003586, A237442, A276380.
%K A348599 tabf,nonn
%O A348599 1,2
%A A348599 _Michael De Vlieger_, Feb 23 2022