A359035 a(n+1) is the smallest number not already used which can be written as the product of two numbers with the same difference as a(n) and a(n-1); a(1)=1 and a(2)=2.

1, 2, 6, 5, 12, 8, 21, 14, 18, 32, 15, 38, 24, 51, 28, 50, 23, 58, 36, 48, 13, 74, 62, 45, 60, 16, 92, 77, 34, 44, 11, 70, 122, 53, 142, 90, 108, 19, 182, 164, 40, 125, 86, 82, 96, 72, 25, 98, 150, 165, 54, 112, 59, 110, 52, 120, 69, 106, 78, 29, 102, 228, 127, 206, 80, 256, 177, 162

Offset: 1

Author

Neal Gersh Tolunsky, Dec 12 2022

Keywords

• nonn

Comments

This sequence starts chaotic but makes a sudden transition into a linear recurrence for n > 433. For details see the formula section. - Thomas Scheuerle, Dec 13 2022

Examples

			a(3) is 6: The difference between the two previous terms is 1 and 2*3=6 is the smallest number that is the product of two numbers with a difference of 1.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..600
• Thomas Scheuerle, Scatter plot log(a(n)) for n = 1...650

Crossrefs

Cf. A002378.

Programs

MATLAB

function a = A359035( max_n )
    a = [1 2];
    for n = 3:max_n
        m = 1; d = abs(a(n-1)-a(n-2));
        while ~isempty(find(a==(m*(m+d)), 1))
            m = m+1;
        end
        a(n) = m*(m+d);
    end
end % Thomas Scheuerle, Dec 13 2022

Formula

From Thomas Scheuerle, Dec 13 2022: (Start)

a(534+(k+1)*3) = 2*a(534+k*3)-1 for k >= 0 and a(534) = 555.

a(535+(k+1)*3) = 2*a(535+k*3)+1 for k >= 0 and a(535) = 2214.

a(536+(k+1)*3) = 2*a(536+k*3)+2 for k >= 0 and a(536) = 3322. (End)