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A277070 Row length of A276380(n). 3
1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n) represents the partition size generated by greedy algorithm at A276380(n) such that all parts k are unique and in A003586.

See A276380 for further comments about the greedy algorithm.

Row n = 1 for n that are in A003586.

A237442(n) represents the smallest possible partition size such that all k are distinct and in A003586. The reference defines the "canonic" representation of n in the "dual-base number system", i.e., base(2,3), essentially as those which have length A237442(n).

a(n) differs from A237442(n) 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. 35-39.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

EXAMPLE

a(n) Terms k in row n of A276380:

1    1

1    2

1    3

1    4

2    1,4

1    6

2    1,6

1    8

1    9

2    1,9

2    2,9

1    12

2    1,12

2    2,12

2    3,12

1    16

2    1,16

1    18

2    1,18

2    2,18

2    3,18

2    4,18

3    1,4,18

...

a(41) = 3 since A276380(41) = {1,4,36}, but {9,32} is the shortest possible partition of 41 such that all terms are distinct and in A003586.

a(88) = 3 since A276380(88) = {1,6,81}, but {16,72} and {24,64} are shorter and have A237442(88) = 2 terms.

MATHEMATICA

Table[Length@ 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, 100}]

CROSSREFS

Cf. A003586, A237442, A276380, A277071.

Sequence in context: A094840 A035218 A237442 * A139355 A039736 A093921

Adjacent sequences:  A277067 A277068 A277069 * A277071 A277072 A277073

KEYWORD

nonn

AUTHOR

Michael De Vlieger, Sep 27 2016

STATUS

approved

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Last modified November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)