1,2

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).

These numbers represent an extreme representation where the digits tend to be placed close together.

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

Michael De Vlieger, Table of n, a(n) for n = 1..10093 (rows n = 1..3600, flattened)

Michael De Vlieger, Plot of parts in row n at (T(n,k), n), for n = 1..256.

Michael De Vlieger, Plot T(n,k) at (T(n,k), n) for n = 1..10000.

Michael De Vlieger, Annotated plot of m = A347860(n,k) and m = T(n,k) at (m, n) 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.

A237442(n) = length of row n.

Triangle begins:

1;

2;

3;

4;

3, 2; (product larger than (4,1))

6;

4, 3; (product larger than (6,1))

8;

9;

6, 4; (product greatest of {(9,1), (8,2), (6,4)})

8, 3; (product larger than (9,2))

12;

...

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

Cf. A003586, A237442, A276380.

Sequence in context: A303597 A251102 A111880 * A306461 A101497 A274007

Adjacent sequences: A348596 A348597 A348598 * A348600 A348601 A348602

tabf,nonn

Michael De Vlieger, Feb 23 2022

approved