A347860 - OEIS

%I #20 Mar 04 2022 11:20:02

%S 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 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 3,36,36,1,36,2,36,3,36,4,32,9,36,6,27,16,36,8,36,9

%N 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 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 These numbers represent an extreme representation where the digits tend to be spread farthest apart in the plot described above.

%C Additionally, this canonic representation of n is often identical to the greedy representation of n shown by row n in A276380.

%D Vassil Dimitrov, Graham Jullien, and Roberto Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press (2012), 35-39.

%H Michael De Vlieger, <a href="/A347860/b347860.txt">Table of n, a(n) for n = 1..10093</a> (rows n = 1..3600, flattened)

%H 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 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 Michael De Vlieger, <a href="/A347860/a347860_2.png">Plot T(n,k) at (T(n,k), n)</a> for n = 1..10000.

%H 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 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 A237442(n) = length of row n.

%e Triangle begins:

%e    1;

%e    2;

%e    3;

%e    4;

%e    4, 1;   (product smaller than (3,2))

%e    6;

%e    6, 1;   (product smaller than (4,3))

%e    8;

%e    9;

%e    9, 1;   (product least of {(9,1), (8,2), (6,4)})

%e    9, 2;   (product smaller than (8,3))

%e   12;

%e   ...

%t nn = 45; s = 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 Cf. A003586, A237442, A276380.

%K tabf,nonn

%O 1,2

%A _Michael De Vlieger_, Feb 23 2022