  The electron correlations near the half-filling of the lowest and excited
Landau levels (LL's) are studied using numerical diagonalization. It is shown
that in the low lying states electrons avoid pair states with relative angular
momenta ${\cal R}$ corresponding to positive anharmonicity of the interaction
pseudopotential $V({\cal R})$. In the lowest LL, the super-harmonic behavior of
$V({\cal R})$ causes Laughlin correlations (avoiding pairs with ${\cal R}=1$)
and the Laughlin-Jain series of incompressible ground states. In the first
excited LL, $V({\cal R})$ is harmonic at short range and a different series of
incompressible states results. Similar correlations occur in the paired
Moore-Read $\nu={5\over2}$ state and in the $\nu={7\over3}$ and ${8\over3}$
states, all having small total parentage from ${\cal R}=1$ and 3 and large
parentage from ${\cal R}=5$. The $\nu={7\over3}$ and ${8\over3}$ states are
different from Laughlin $\nu={1\over3}$ and ${2\over3}$ states and, in finite
systems, occur at a different LL degeneracy (flux). The series of Laughlin
correlated states of electron pairs at $\nu=2+2/(q_2+2)={8\over3}$,
${5\over2}$, ${12\over5}$, and ${7\over3}$ is proposed, although only in the
$\nu={5\over2}$ state pairing has been confirmed numerically. In the second
excited LL, $V({\cal R})$ is sub-harmonic at short range and (near the
half-filling) the electrons group into spatially separated larger $\nu=1$
droplets to minimize the number of strongly repulsive pair states at ${\cal
R}=3$ and 5.
