A348599 -id:A348599 - cp's OEIS frontend

A276380 Irregular triangle where row n contains terms k of the partition of n produced by greedy algorithm such that all elements are in A003586.

1, 2, 3, 4, 1, 4, 6, 1, 6, 8, 9, 1, 9, 2, 9, 12, 1, 12, 2, 12, 3, 12, 16, 1, 16, 18, 1, 18, 2, 18, 3, 18, 4, 18, 1, 4, 18, 24, 1, 24, 2, 24, 27, 1, 27, 2, 27, 3, 27, 4, 27, 32, 1, 32, 2, 32, 3, 32, 36, 1, 36, 2, 36, 3, 36, 4, 36, 1, 4, 36, 6, 36, 1, 6, 36, 8, 36, 9, 36, 1, 9, 36, 2, 9, 36, 48, 1, 48

Offset: 1

Michael De Vlieger, Sep 25 2016

This sequence uses a greedy algorithm f(x) to find the largest number k <= n such that k is in A003586. The function is recursively applied to the result until it reaches 1. This is the algorithm described in the reference p. 36. This sequence presents the terms in order from least to greatest term.
The reference suggests the greedy algorithm is one way to render n in a "dual-base number system", essentially base (2,3) with bases 2 and 3 arranged orthogonally to produce a matrix of places with values that are the tensor product of prime power ranges of 2 and 3. Place values are signified by 0 or 1. Thus we can boil down the matrix to simply list the values of places harboring digit 1.
Row n = n for n that are in A003586.
The reference defines a "canonic" representation of n on page 33 as having the lowest number of terms. The greedy algorithm does not always render the canonic representation. a(41) = {1,4,36}, but {9,32} is the shortest possible partition of 41 such that all terms are in A003586.
The terms in row n differ from the canonic terms at n = 41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, 185, 209, 215, 226, 236, 239, 248... (i.e., A277071).

			Triangle begins:
1
2
3
4
1,4
6
1,6
8
9
1,9
2,9
12
1,12
2,12
3,12
16
1,16
18
1,18
2,18
3,18
4,18
1,4,18
...

V. Dimitrov, G. Jullien, and R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.

Michael De Vlieger, Table of n, a(n) for n = 1..11006 (Rows 1 <= n <= 3600)

Cf. A003586, A237442 (least number of 3-smooth numbers that add up to n), A277070 (row lengths), A277071, A347860, A348599.

Table[Reverse@ DeleteCases[Append[Abs@ Differences@ #, Last@ #], k_ /; k == 0] &@ NestWhileList[# - SelectFirst[# - Range[0, # - 1], Block[{m = #, n = 6}, While[And[m != 1, ! CoprimeQ[m, n]], n = GCD[m, n]; m = m/n]; m == 1] &] &, n, # > 1 &], {n, 49}]

from itertools import count, takewhile
N = 50
def B(p): return list(takewhile(lambda x: x<=N, (p**i for i in count(0))))
B23set = set(b*t for b in B(2) for t in B(3) if b*t <= N)
B23lst = sorted(B23set, reverse=True)
def row(n):
    if n in B23set: return [n]
    big = next(t for t in B23lst if t <= n)
    return row(n - big) + [big]
print([t for r in range(1, N) for t in row(r)]) # Michael S. Branicky, Sep 14 2022

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.

Original entry on oeis.org

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

Michael De Vlieger, Feb 23 2022

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.

			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;
  ...

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, 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.

Cf. A003586, A237442, A276380.

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

A237442(n) = length of row n.

cp's OEIS Frontend

A276380 Irregular triangle where row n contains terms k of the partition of n produced by greedy algorithm such that all elements are in A003586.

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