How many photons does an orbiting electron send per a cycle?

Let us have an electron moving in a circular orbit at a constant velocity. How many photons does it radiate per cycle?

https://en.wikipedia.org/wiki/Larmor_formula

For a relativistic electron, the radiated power grows as γ⁴, where γ is the Lorentz factor.

Let us calculate with the non-relativistic Larmor formula. The acceleration is

       a = v² / r

          = ω² r.

According to the Larmor formula, the radiated power is

       P = e² ω⁴ r² / (6 π ε₀ c³).

The cycle time is t = 2 π / ω.

The energy per a single cycle is

       E = e² ω³ r² / (3 ε₀ c³). 

The energy of a single photon is

       E' = h f = h ω / (2 π).

The number of photons per cycle is

       n = 2/3 π e² ω² r² / (h ε₀ c³)

          = 2/3 π e² v² / (h ε₀ c³)

          = 0.031 * v² / c².

That is, if the electron is mildly relativistic (v ~ c), it takes ~ 33 cycles to emit one photon. The number 33 is not very far from 1. Is it a coincidence that a relatively small number of cycles is able to produce a single photon?

If the velocity v = 0.01 c, like for the electron in the hydrogen atom, it takes 330,000 cycles to produce a photon.

The energy of a single photon for a mildly relativistic electron is

       E'' = h ω / (2 π)

            ~ h c / (2 π r)

            = h / t,

where t is the cycle time. This is the energy - time uncertainty relation:

       ΔE Δt ≥ h.

We can derive the uncertainty relation by building a wave packet where we assume that the photon energy of a wave of a frequency f is h f.