Produce a set A such that r(n)>0 for all n in [1,N], but with |A|<= sqrt(4N+1). Note that r(n)=∣∣{(a,a′):a,a′ in A,n=a+a′}∣∣

Cindy Noble 2022-09-04 Answered
Produce a set A such that r ( n ) > 0 for all n [ 1 , N ], but with | A | 4 N + 1 .
Note that
r ( n ) = | { ( a , a ) : a , a A , n = a + a } |
A = { 0 , 1 , 2 } would work with the interval being [ 1 , 4 ]. Then 3 17 .
A second part of the question shows that one can prove that | A | N if it satisfies the above conditions. But 3 > 4 = 2. Does this mean that my set A is wrong?
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Answers (1)

Answered 2022-09-05 Author has 9 answers
It should be | A | N , since you need at least that many numbers in A to form enough pairs to produce all the N numbers.
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