A390080 Even numbers k such that Sum_{i=1..r} i*(k-i) = Sum_{j=1..s} j*(j+k) for some integers r and s.

14, 24, 28, 42, 48, 50, 52, 54, 62, 70, 72, 76, 80, 90, 92, 94, 100, 102, 106, 110, 114, 116, 124, 128, 132, 134, 138, 140, 154, 158, 166, 172, 174, 176, 178, 180, 184, 192, 198, 200, 202, 204, 206, 210, 212, 214, 224, 226, 230, 232, 234, 236, 242, 248, 252

Offset

1,1

Comments

Place k at the fulcrum of a massless seesaw. Then place the next r smaller positive numbers to the left and the next s larger numbers to the right. Suppose that each number creates a torque about the fulcrum equal to the product of the number and its distance from the fulcrum. The number k is in the sequence if r and s can be found to balance the seesaw; that is, the torque on the left balances the torque on the right.

This sequence extends the idea of A094550 by accounting for the distance from each number to the fulcrum.

Terms in the sequence satisfy 2*s^2 + (1+3*k-2*r)*s + r*(1-3*k+2*r) = 0 for some r and s.

All odd numbers satisfy the equation above: if k = 2*t+1, then a solution is r = 2*t and s = t.

Links

Chris R. Rehmann, Table of n, a(n) for n = 1..10000

Example

14 is a term because the numbers 4 through 13 create a total torque of 385 units on the left and the numbers 15 through 20 create an opposing torque of 385 units on the right.

Prog

(MATLAB) isOK = @(k) any(mod(arrayfun(@(r) -1-3*k+2*r+sqrt(1+9*k^2-12*r.*(1+r)+6*k*(1+2*r)), 1:k-1), 4)==0);

Crossrefs

Cf. A094550.

Sequence in context: A020905 A048635 A114872 * A106699 A111455 A171124

Adjacent sequences: A390077 A390078 A390079 * A390081 A390082 A390083

Keyword

nonn

Author

Chris R. Rehmann, Dec 28 2025

Status

approved