Consider the finite sums

$$ F_n (q) = sum_ {k = 1} ^ nq ^ { binom {k} 2} $$

with exponents the *triangular numbers* $ binom {k} $ 2, When $ n $ strange, it seems that $ F_n (q) $ does not factor over $ mathbb {Z} (q) $, On the other hand, if $ n = $ 2 million is just

**QUESTION.** is it true that $ F_ {2m} (q) $ is divisible by the product

$$ prod_ {j geq0} (1 + q ^ {m / 2 ^ j}) $$

where the product lasts as long as $ m / 2 ^ j $ is an integer.

**Examples.** Here is an example:

begin {align}

(1 + q ^ 2) (1 + q) , vert & , F_4 (q); qquad (1 + q ^ 3) , , vert , F_6 (q); \

(1 + q ^ 4) (1 + q ^ 2) (1 + q) , , vert & , F_8 (q); qquad (1 + q ^ 6) (1 + q ^ 3) , , vert , F_ {12} (q).

end