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A225179 a(n) = min{2 + c_n + 2(c_1 + c_2 + ... + c_(n-1)) | n = p + q, 1 <= p < q < n, gcd(p, q) = 1, and p/q has continued fraction expansion [0; c_1, c_2, ...., c_n]}. 1

%I #23 May 03 2013 04:49:24

%S 1,3,4,5,6,7,7,8,8,9,9,10,10,10,11,11,11,11,12,12,12,12,12,13,13,13,

%T 13,13,13,14,14,14,14,14,14,15,14,15,15,15,15,16,15,15,15,16,15,16,16,

%U 16,16,16,16,17,16,17,16,17,17,16,17,17,17,17,17,18,17,17,17,17,17,19,17,18,18,17,18,18,18,18,18,19,18

%N a(n) = min{2 + c_n + 2(c_1 + c_2 + ... + c_(n-1)) | n = p + q, 1 <= p < q < n, gcd(p, q) = 1, and p/q has continued fraction expansion [0; c_1, c_2, ...., c_n]}.

%C It is conjectured that a(n) = min{ B(w) | w is unbordered and length(w) = n }, where B(w) is the number of distinct unbordered factors in w.

%C If this conjecture were to be proved, we could use this is the definition, and put the present definition into the FORMULA field. It is known to be true for n <= 20.

%C [A word v is bordered if there is a nonempty word u different from v which is both a prefix and a suffix of v (see, for example, Allouche and Shallit).]

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 28.

%D Kalle Saari, Open problem presented at Workshop on Challenges in Combinatorics on Words, Fields Institute, Toronto, April 21-26, 2013.

%H Kalle Saari, <a href="/A225179/b225179.txt">Table of n, a(n) for n = 1..199</a>

%H Kalle Saari, <a href="http://users.utu.fi/kasaar/unbordered.txt">Unbordered words with the smallest number of distinct unbordered factors</a>

%K nonn

%O 1,2

%A _N. J. A. Sloane_, May 03 2013, based on an email from Kalle Saari, Apr 23 2013

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