The electron correlations near the half-filling of the lowest and excited L
andau levels (LL's) are studied using numerical diagonalization. It is show
n that in the low-lying states electrons avoid pair states with relative an
gular momenta R corresponding to positive anharmonicity of the interaction
pseudopotential V (R). In the lowest LL, the superharmonic behavior of V(R)
causes Laughlin correlations (avoiding pairs with R = 1) and the Laughlin-
Jain series of incompressible ground states. In the first excited LL, V(R)
is harmonic at short range and a different series of incompressible states
results. Similar correlations occur in the paired Moore-Read v = 5/2 state
and in the v = 7/3 and 8/3 states, all having small total parentage from R
= 1 and 3 large parentage from R = 5. The v = 7/3 and 8/3 states are differ
ent from Laughlin v = 1/3 and 2/3 states and, in finite systems, occur at a
different LL degeneracy (flux). The series of Laughlin-correlated states o
f electron pairs at v = 2 + 2/(q(2) + 2) = 8/3, 5/2, 12/5, and 7/3 is propo
sed, although only in the v = 5/2 state pairing has been confirmed numerica
lly. In the second excited LL, V(R) is subharmonic at short range and (near
the half-filling) the electrons group into spatially separated larger v =
1 droplets to minimize the number of strongly repulsive pair states at R =
3 and 5.