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A347860 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 minimal.

%I A347860 #23 Jun 20 2025 18:49:50
%S A347860 1,2,3,4,4,1,6,6,1,8,9,9,1,9,2,12,12,1,12,2,12,3,16,16,1,18,18,1,18,2,
%T A347860 18,3,18,4,18,4,1,24,24,1,24,2,27,27,1,27,2,27,3,27,4,32,32,1,32,2,32,
%U A347860 3,36,36,1,36,2,36,3,36,4,32,9,36,6,27,16,36,8,36,9
%N A347860 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 minimal.
%C A347860 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 A347860 These numbers represent an extreme representation where the digits tend to be spread farthest apart in the plot described above.
%C A347860 Additionally, this canonic representation of n is often identical to the greedy representation of n shown by row n in A276380.
%D A347860 Vassil Dimitrov, Graham Jullien, and Roberto Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press (2012), 35-39.
%H A347860 Michael De Vlieger, <a href="/A347860/b347860.txt">Table of n, a(n) for n = 1..10093</a> (rows n = 1..3600, flattened)
%H A347860 Michael De Vlieger, <a href="/A347860/a347860.png">Plot of parts in row n at (T(n,k), n)</a> for n = 1..256.
%H A347860 Michael De Vlieger, <a href="/A347860/a347860_1.png">Comparison of row n of this sequence with row n of A276380</a> for n = 1..256, showing terms of this sequence in blue, and those of A276380 in red. Where these coincide, we plot in black.
%H A347860 Michael De Vlieger, <a href="/A347860/a347860_2.png">Plot T(n,k) at (T(n,k), n)</a> for n = 1..10000.
%H A347860 Michael De Vlieger, <a href="/A347860/a347860_3.png">Annotated plot of T(n,k) and S(n,k) = A276380(n,k)</a>, n = 1..128, accentuating T(n,k) in blue and S(n,k) in red, otherwise in black and white where they coincide. S(n,k) is the result of a greedy algorithm described in Dimitrov, et al., i.e., more parts such that the row sum equals n.
%H A347860 Michael De Vlieger, <a href="/A347860/a347860_4.png">Annotated plot of m = A348599(n,k) and m = T(n,k) at (m, n)</a> for n = 1..64, showing m in row n of this sequence in red, m in row n of A347860 in blue, but in black if these coincide.
%F A347860 A237442(n) = length of row n.
%e A347860 Triangle begins:
%e A347860    1;
%e A347860    2;
%e A347860    3;
%e A347860    4;
%e A347860    4, 1;   (product smaller than (3,2))
%e A347860    6;
%e A347860    6, 1;   (product smaller than (4,3))
%e A347860    8;
%e A347860    9;
%e A347860    9, 1;   (product least of {(9,1), (8,2), (6,4)})
%e A347860    9, 2;   (product smaller than (8,3))
%e A347860   12;
%e A347860   ...
%t A347860 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++]; MinimalBy[w, Times @@ # &][[1]]], {n, nn}] // Flatten
%Y A347860 Cf. A003586, A237442, A276380.
%K A347860 tabf,nonn
%O A347860 1,2
%A A347860 _Michael De Vlieger_, Feb 23 2022