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.

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

Offset: 1

Author

Michael De Vlieger, Feb 23 2022

Keywords

• tabf
• nonn

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.

Examples

			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;
  ...
		

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.

Crossrefs

Cf. A003586, A237442, A276380.

Programs

Mathematica

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

Formula

A237442(n) = length of row n.