login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
REFERENCES
V. Dimitrov, G. Jullien, and R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11006 (Rows 1 <= n <= 3600)
EXAMPLE
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
...
MATHEMATICA
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}]
PROG
(Python)
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
CROSSREFS
Cf. A003586, A237442 (least number of 3-smooth numbers that add up to n), A277070 (row lengths), A277071, A347860, A348599.
Sequence in context: A257053 A129717 A317088 * A352724 A248723 A117742
KEYWORD
nonn,tabf,easy
AUTHOR
Michael De Vlieger, Sep 25 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)