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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.

%I A361942 #5 Apr 01 2023 08:54:12
%S A361942 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,
%T A361942 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,
%U A361942 4,6,5,6,7,3,5,6,4,6,5,6,7,6,5,6,7,2,5
%N 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.
%C A361942 Leading zeros in binary expansions of positive integers are ignored.
%C A361942 This sequence is a variant of A302291 related to fractional powers of words.
%C A361942 For any k > 0, the value k appears A045690(k) times in a(2^(k-1)), ..., a(2^k-1).
%D A361942 Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 23.
%H A361942 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A361942 a(n) <= A302291(n).
%F A361942 a(n) <= A070939(n) with equality iff n belongs to A091065.
%F A361942 a(2^k-1) = 1 for any k >= 0.
%F A361942 a(2^k) = k+1 for any k >= 0.
%e A361942 The first terms, alongside the binary expansion of n split into chunks of length a(n), are:
%e A361942   n   a(n)  bin(n)
%e A361942   --  ----  ------
%e A361942    0     1  0
%e A361942    1     1  1
%e A361942    2     2  10
%e A361942    3     1  1|1
%e A361942    4     3  100
%e A361942    5     2  10|1
%e A361942    6     3  110
%e A361942    7     1  1|1|1
%e A361942    8     4  1000
%e A361942    9     3  100|1
%e A361942   10     2  10|10
%e A361942   11     3  101|1
%e A361942   12     4  1100
%e A361942   13     3  110|1
%e A361942   14     4  1110
%e A361942   15     1  1|1|1|1
%o A361942 (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););); }
%Y A361942 Cf. A045690, A070939, A091065, A302291.
%K A361942 nonn,base
%O A361942 0,3
%A A361942 _Rémy Sigrist_, Mar 31 2023