A361942 - OEIS

A361942

For any number n >= 0 with binary expansion (b_1, ..., b_w), a(n) is the least p > 0 such that b_i = b_{p+i} for i = 1..w-p.

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 2, 3, 4, 3, 4, 1, 5, 4, 3, 4, 5, 2, 3, 4, 5, 4, 5, 3, 5, 4, 5, 1, 6, 5, 4, 5, 3, 5, 4, 5, 6, 5, 2, 5, 6, 3, 4, 5, 6, 5, 6, 4, 6, 5, 3, 4, 6, 5, 6, 4, 6, 5, 6, 1, 7, 6, 5, 6, 4, 6, 5, 6, 7, 3, 5, 6, 4, 6, 5, 6, 7, 6, 5, 6, 7, 2, 5

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OFFSET

0,3

COMMENTS

Leading zeros in binary expansions of positive integers are ignored.

This sequence is a variant of A302291 related to fractional powers of words.

For any k > 0, the value k appears A045690(k) times in a(2^(k-1)), ..., a(2^k-1).

REFERENCES

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 23.

LINKS

Table of n, a(n) for n=0..86.

Index entries for sequences related to binary expansion of n

FORMULA

a(n) <= A302291(n).

a(n) <= A070939(n) with equality iff n belongs to A091065.

a(2^k-1) = 1 for any k >= 0.

a(2^k) = k+1 for any k >= 0.

EXAMPLE

The first terms, alongside the binary expansion of n split into chunks of length a(n), are:

n a(n) bin(n)

-- ---- ------

0 1 0

1 1 1

2 2 10

3 1 1|1

4 3 100

5 2 10|1

6 3 110

7 1 1|1|1

8 4 1000

9 3 100|1

10 2 10|10

11 3 101|1

12 4 1100

13 3 110|1

14 4 1110

15 1 1|1|1|1

PROG

(PARI) a(n) = { my (b = if (n, binary(n), [0])); for (p = 1, oo, if (b[1..#b-p] == b[1+p..#b], return (p); ); ); }

CROSSREFS

Cf. A045690, A070939, A091065, A302291.

Sequence in context: A372917 A057432 A374741 * A302295 A215467 A284266

Adjacent sequences: A361939 A361940 A361941 * A361943 A361944 A361945

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Mar 31 2023

STATUS

approved