A277070 -id:A277070 - cp's OEIS frontend

A277071 Numbers n for which A277070(n) does not equal A237442(n).

41, 43, 59, 86, 88, 91, 113, 118, 123, 135, 155, 172, 176, 177, 182, 185, 209, 215, 226, 236, 239, 248, 261, 267, 270, 273, 275, 279, 307, 310, 311, 337, 339, 344, 347, 352, 354, 364, 365, 367, 369, 370, 371, 377, 383, 405, 407, 418, 425, 427, 430, 452, 455, 465, 472, 473, 475, 478, 479, 496, 499

Offset: 1

Michael De Vlieger, Sep 27 2016

nonn

These are numbers n for which the greedy algorithm A276380(n) produces a partition of n with more than A237442(n) terms that are all unique and in A003586.
A276380(n) = A237442(n) if n is in A003586. There may be more than one partition of n that has terms that are unique and in A003586. The first n in a(n) with that quality is n = 88.
A277070(n)-A237442(n) = 1 at {41, 43, 59, 86, 88, 91, 113, 118, ...}
A277070(n)-A237442(n) = 2 at {279, 371, 558, 837, 1116, 1240, 1267, ...}
A277070(n)-A237442(n) = 3 at {2777, 5554, ...}

			41 is in the sequence because A276380(41) = {1,4,36}, thus A277070(41) = 3, but A237442(41) = 2. The partition of 41 with unique terms that are all in A003586 is {9,32}.
88 is in the sequence because A276380(88) = {1,6,81}, thus A277070(88) = 3, but A237442(41) = 2. There are 2 partitions of 88 with unique terms that are all in A003586: {16,72} and {24,64}.

V. Dimitrov, G. Jullien, 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..3000

Cf. A003586, A237442, A276380, A277070.

f[n_] := Length@ DeleteCases[Append[Abs@ Differences@ #, Last@ #], k_ /; k == 0] &@ NestWhileList[# - SelectFirst[# - Range[0, # - 1], Module[{a = #, b = 6}, While[And[a != 1, ! CoprimeQ[a, b]], b = GCD[a, b]; a = a/b]; a == 1] &] &, n, # > 1 &]; g[n_] := Block[{p = Select[Range@ n, FactorInteger[#][[-1, 1]] < 4 &], k = 1}, While[{} == Quiet@ IntegerPartitions[n, {k}, p, 1], k++]; k]; Select[Range@ 500, f@ # != g@ # &] (* function g after Giovanni Resta at A237442 *)

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.

Original entry on oeis.org

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

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A277071 Numbers n for which A277070(n) does not equal A237442(n).

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