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A283297
The smallest cardinality of a difference-basis in the cyclic group of order n.
0
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, 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, 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
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.
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
Sequence in context: A331246 A331256 A368941 * A091194 A156079 A268708
KEYWORD
nonn
AUTHOR
Taras Banakh, Mar 04 2017
STATUS
approved