A293843 Split the infinite binary word A030302, from left to right, into the largest possible cubefree chunks; a(n) = length of n-th cubefree chunk.

5, 7, 4, 4, 10, 3, 9, 5, 2, 3, 5, 12, 9, 10, 2, 3, 7, 9, 2, 5, 4, 2, 4, 2, 4, 2, 4, 6, 6, 3, 8, 15, 6, 12, 5, 2, 14, 2, 6, 2, 4, 6, 3, 11, 13, 8, 2, 2, 4, 3, 5, 7, 4, 2, 6, 5, 2, 5, 2, 2, 3, 2, 5, 2, 5, 7, 4, 3, 7, 7, 7, 3, 7, 15, 7, 19, 7, 2, 5, 7, 11, 5, 5

Offset: 1

Author

Rémy Sigrist, Oct 17 2017

Keywords

• nonn
• base

Comments

Using A030190 instead of A030302 leads to the same sequence except for the first term (that would equal 6).

The word A030302 contains infinitely many consecutive triples of 0's (corresponding for example to the last binary digits of multiples of 8), hence A030302 has no infinite cubefree suffix, and this sequence is well defined for any n > 0.

a(n) >= 2 for any n > 0.

This sequence is unbounded: for any n > 0:

- A028445(2*n) > 0,

- hence we can choose a number in A286262, say c, with 2*n digits in binary,

- and the subword of A030302 corresponding to c will participate in no more than two cubefree chunks,

- and one of those chunks will have at least length n, QED.

The first records of the sequence are (see also A293867 and A293868):

a(n) n

---- --

5 1

7 2

10 5

12 12

15 32

19 76

21 212

25 412

35 418

36 2305

39 5118

47 5516

59 49014

63 104902

67 261530

71 478638

75 1016483

79 2148745

83 4532050

87 9534639

91 20011894

95 41896466

Examples

			The following table shows the first terms of the sequence, alongside the corresponding cubefree chunks:
n   a(n)  n-th chunk
--  ----  ----------
1   5     11011
2   7          1001011
3   4                 1011
4   4                     1100
5   10                        0100110101
6   3                                   011
7   9                                      110011011
8   5                                               11011
9   2                                                    11
10  3                                                      100
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PERL program for A293843

Crossrefs

Cf. A028445, A030190, A030302, A286262, A293867, A293868.