$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - \dfrac{691}{132}$
$\zeta(-11) = \dfrac{1}{32760}$
$\zeta(-13) = - \dfrac{3617}{12}$
What I am interested in are the sequence of denominators of these fractions.
$12, 120, 252, 240, 132, 32760, 12, 8160, 14364, 6600, 276, 65520, 12, 3480, 85932, 16320, 12, 69090840, 12, 541200, 75852, 2760, 564, 2227680, 132, 6360, 43092, 6960, 708, 3407203800, 12, 32640, 388332, 120, 9372, 10087262640$
They are alternately divisible by 12 and 120.
In the first $1000$ terms, $120$ occurs $53$ times, and $12$ occurs even more often. So it seems they both occur infinitely many times.
My question is: Why? What is the meaning of all this? Does that make $12$ and $120$ some kinds of special numbers?
