

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.


5



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
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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 "dualbase 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, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 3539.


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}]


CROSSREFS

Cf. A003586, A237442 (least number of 3smooth numbers that add up to n), A277070 (row lengths), A277071.
Sequence in context: A257053 A129717 A317088 * A248723 A117742 A117716
Adjacent sequences: A276377 A276378 A276379 * A276381 A276382 A276383


KEYWORD

nonn,tabf,easy


AUTHOR

Michael De Vlieger, Sep 25 2016


STATUS

approved



