OFFSET
1,2
COMMENTS
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.
LINKS
T. Banakh, V. Gavrylkiv, Difference bases in cyclic and dihedral groups, arXiv:1702.02631 [math.CO], 2017.
MathOverflow, What is the smallest cardinality of a self-linked set in a finite cyclic group?, Feb 15 2017.
J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377-85.
FORMULA
a(n) = q+1 if n=1+q+q*q for a power of a prime number.
a(n) >= (1+sqrt(4n-3))/2;
a(n) <= sqrt(2n) for n != 4;
a(n) < sqrt(2n) if n>=5 and sqrt(n/8) is not integer.
It is an open problem whether a(n) = (1+o(1))sqrt(n). See the MathOverflow link.
CROSSREFS
KEYWORD
nonn
AUTHOR
Taras Banakh, Mar 04 2017
STATUS
approved