A357574 a(n) is the maximum number of pairs that sum to a power of 2 in a set of n consecutive odd numbers.

0, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 150, 153, 157, 160, 164, 167

Offset: 1

Author

Thomas Scheuerle, Oct 04 2022

Keywords

• nonn

Comments

An optimal set delivering a(n) pairs summing to powers of 2 can be formed by n-A357409(n) first negative odd numbers and A357409(n) first positive odd numbers, that is, by the odd numbers in the interval [-2*(n-A357409(n))+1, 2*A357409(n)-1].

a(n) is in many cases equal to A347301(n) but there are some deviations: a(3) = 2 but A347301(3) = 3, a(29) = 62 but A347301(29) = 61, a(31) = 68 but A347301(32) = 67, a(33) = 74 but A347301(33) = 73, ... . Hence it appears that a(n) may be used as an improved lower bound for A352178(n) in many cases.

Conjecture: a(n+1) - a(n) = k, if n is even then A129868(k-1) < n < A129868(k), if n is odd then A020515(k) <= n < A020515(k+1).

Examples

			a(5) = 5 because A357409(5) = 4, for which the corresponding set {-1, 1, 3, 5, 7} produces 5 powers of 2: 1+3, 1+7, 3+5, 3-1, 5-1.
		

Links

• Thomas Scheuerle, a(n+1) - a(n) for n = 1..500
• Max A. Alekseyev, On computing sets of integers with maximum number of pairs summing to powers of 2, arXiv:2303.02872 [math.CO], 2023.

Crossrefs

Cf. A020515, A129868, A274089, A347301, A352178, A357409.

Programs

MATLAB

function a = A357574( max_n )
    a(1) = 0; q = [];
    for n = 1:max_n
        c = 0;
        for k = 0:n
            s = (2*([0:n]-k))+1;
            r = countpowtwo(s);
            if c < r
                c = r;
                q = s;
            end
        end
        a(n+1) = c;
    end
end
function c = countpowtwo(s)
    M = repmat(s, [length(s), 1]);
    M = M+M';
    M(M<=0) = 7;
    M = bitand(M, M-1);
    M = M + eye(size(M));
    c = length(find(M == 0))/2;
end

Formula

a(n) <= A352178(n).

a(n) >= n-1. This would be the maximum value that could be attained for a set of only positive odd numbers and size n.

Extensions

Edited by Max Alekseyev, Mar 09 2023