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%I #25 Mar 05 2017 18:36:29
%S 1,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,5,6,6,6,6,6,6,6,7,7,6,7,7,7,
%T 7,7,7,8,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,8,9,9,9,9,9,9,9,9,10,10,
%U 10,10,10,10,10,9,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,10,11,12,12,12,12,12,12,12,12
%N The smallest cardinality of a difference-basis in the cyclic group of order n.
%C A subset B is called a difference-basis for an Abelian group G if B-B=G. This sequence is calculated by computer. For n=1+q+q*q where q is a power of a prime number the smallest cardinality of a difference-basis equals q+1, which is witnessed by the difference set of Singer. The problem of calculating the values of the sequence seems to be of exponential complexity.
%H T. Banakh, V. Gavrylkiv, <a href="https://arxiv.org/abs/1702.02631">Difference bases in cyclic and dihedral groups</a>, arXiv:1702.02631 [math.CO], 2017.
%H MathOverflow, <a href="http://mathoverflow.net/questions/262317">What is the smallest cardinality of a self-linked set in a finite cyclic group?</a>, Feb 15 2017.
%H J. Singer, <a href="http://dx.doi.org/10.1090/S0002-9947-1938-1501951-4">A theorem in finite projective geometry and some applications to number theory</a>, Trans. Amer. Math. Soc. 43 (1938), 377-85.
%F a(n) = q+1 if n=1+q+q*q for a power of a prime number.
%F a(n) >= (1+sqrt(4n-3))/2;
%F a(n) <= sqrt(2n) for n != 4;
%F a(n) < sqrt(2n) if n>=5 and sqrt(n/8) is not integer.
%F It is an open problem whether a(n) = (1+o(1))sqrt(n). See the MathOverflow link.
%K nonn
%O 1,2
%A _Taras Banakh_, Mar 04 2017