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A211100 Number of factors in Lyndon factorization of binary expansion of n. 6
1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 2, 3, 2, 6, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 7, 3, 4, 3, 5, 3, 4, 3, 6, 4, 5, 3, 5, 4, 4, 3, 7, 4, 5, 4, 6, 5, 5, 4, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). a(n) = number of factors in Lyndon factorization of binary expansion of n.

It appears that a(n) = k for the first time when n = 2^(k-1)+1.

REFERENCES

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.

G. Melancon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Maple programs for A211100 etc.

EXAMPLE

n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors, so a(25) = 3.

Here are the Lyndon factorizations for small values of n:

.0.

.1.

.1.0.

.1.1.

.1.0.0.

.1.01.

.1.1.0.

.1.1.1.

.1.0.0.0.

.1.001.

.1.01.0.

.1.011.

.1.1.0.0.

...

CROSSREFS

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof).

A211095 and A211096 give information about the smallest (or rightmost) factor. Cf. A211097, A211098, A211099.

Sequence in context: A217865 A185166 A276555 * A105264 A063787 A182745

Adjacent sequences:  A211097 A211098 A211099 * A211101 A211102 A211103

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mar 31 2012

STATUS

approved

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Last modified February 19 20:17 EST 2019. Contains 320328 sequences. (Running on oeis4.)