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More protons in the nucleus attract the electrons more strongly. As a consequence, the orbitals get smaller too. The combination of both lets electrons with the same orbital numbers have a binding energy increase like the square of the number of protons more or less. So you won't find the same transitions (like 3d->3p) at the same wavelength. So, many visible hydrogen lines land deep in the ultraviolet for atoms with more protons.
I wrote "more or less" like the square because all electrons interact, especially those on the same orbital. The square law works for one single electron around nuclei of varied numbers of protons.
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Electrons emit light when "falling" from a orbital where they were to an orbital where a place is vacant. For instance, when the 1s shell of lithium is full, as is usual, the two electrons can't fall deeper, and no electron can fall on this filled orbital, so the 1s doesn't contribute to the spectrum. And since usual spectra on Earth use to deal with only one excited electron, which is already rare enough, the intensity doesn't increase like the number of electrons.
The intensity of a line depends on the probability that the excited electron takes the related de-excitation transition rather than other transitions.
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The number of lines depends on the orbitals available as a start and as a finish for the electron(s). More electrons in an atoms don't change radically this number, and especially, they don't create more possible states for the electrons proportionally to the number of electrons present. The effects are more subtle: for instance more electrons fill more states, which hence aren't available as a finish for an electronic transition. Also, the interactions between the electrons generally separate the energies of different states that have one single energy with one electron only. For instance 2s and 2p have (almost exactly) the same energy in hydrogen but different ones in carbon. Or already with helium, the possible energies differ (a lot) when the spins of both electrons are parallel or antiparallel.
So the general picture is a mess, and you can't expect the number of lines to increase simply with the number of electrons. It depends on which wavelength range you observe, and whether many lines (their number is infinite) fall in that range.