OFFSET
1,2
COMMENTS
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 spread farthest apart in the plot described above.
Additionally, this canonic representation of n is often identical to the greedy representation of n shown by row n in A276380.
REFERENCES
Vassil Dimitrov, Graham Jullien, and Roberto Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press (2012), 35-39.
LINKS
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, Comparison of row n of this sequence with row n of A276380 for n = 1..256, showing terms of this sequence in blue, and those of A276380 in red. Where these coincide, we plot in black.
Michael De Vlieger, Plot T(n,k) at (T(n,k), n) for n = 1..10000.
Michael De Vlieger, Annotated plot of T(n,k) and S(n,k) = A276380(n,k), 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.
Michael De Vlieger, Annotated plot of m = A348599(n,k) and m = T(n,k) at (m, n) 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.
FORMULA
A237442(n) = length of row n.
EXAMPLE
Triangle begins:
1;
2;
3;
4;
4, 1; (product smaller than (3,2))
6;
6, 1; (product smaller than (4,3))
8;
9;
9, 1; (product least of {(9,1), (8,2), (6,4)})
9, 2; (product smaller than (8,3))
12;
...
MATHEMATICA
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
CROSSREFS
KEYWORD
tabf,nonn
AUTHOR
Michael De Vlieger, Feb 23 2022
STATUS
approved