User blog comment:Hyp cos/Catching Function - Better Definition/@comment-5529393-20140703172816

It looks to me that B(a+1, b) is approximately phi(a, b).

B(1, b) = b*2

B(2, b) = b*2^b

B(3, b) is b applications of b*2^b, so B(3, w(1+b)) ~ epsilon_b, except there is an offset at fixed points ( B(3, zeta_b) is not zeta_b, it's epsilon_{zeta_b * 2}, so there is an offset until B(3, zeta_b * w), which I believe is epsilon_{zeta_b * w} )

B(4, w(1+b)) ~ zeta_b, except for the offset at fixed points again.

B(n+2, w(1+b)) ~ phi(n, b)

B(w, b) = sup B(n, b) = the smallest ordinal of the form phi(w, a) that is greater than b.

B(w+1, b) = the bth ordinal of the form phi(w, a) greater than b, which be phi(w, b), except for the offset at fixed points again.

B(w+2, w(1+b)) ~ phi(w+1, b)

B(w+n+1, w(1+b)) ~ phi(w+n, b)

B(w2, b) = the smallest ordinal of the form phi(w2, a) that is greater than b.

B(w2+1, b) ~ phi(w2, b)

B(w2+n+1, w(1+b)) ~ phi(w2+n, b)

As far as I can see, this pattern continues forever. So the B function closely follows the phi function, and C(a) = Gamma_a.