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A361942
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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.
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1
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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
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0,3
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COMMENTS
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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).
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REFERENCES
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Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 23.
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LINKS
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FORMULA
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a(2^k-1) = 1 for any k >= 0.
a(2^k) = k+1 for any k >= 0.
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EXAMPLE
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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
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PROG
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(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); ); ); }
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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