A390080 - OEIS

%I #38 Jan 13 2026 03:06:22

%S 14,24,28,42,48,50,52,54,62,70,72,76,80,90,92,94,100,102,106,110,114,

%T 116,124,128,132,134,138,140,154,158,166,172,174,176,178,180,184,192,

%U 198,200,202,204,206,210,212,214,224,226,230,232,234,236,242,248,252

%N 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.

%C 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.

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

%C 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.

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

%H Chris R. Rehmann, <a href="/A390080/b390080.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%o (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);

%Y Cf. A094550.

%K nonn

%O 1,1

%A _Chris R. Rehmann_, Dec 28 2025