  For positive integer p=k+2, we construct a logarithmic extension of the
^sl(2)_k conformal field theory of integrable representations by taking the
kernel of two fermionic screening operators in a three-boson realization of
^sl(2)_k. The currents W^-(z) and W^+(z) of a W-algebra acting in the kernel
are determined by a highest-weight state of dimension 4p-2 and charge 2p-1, and
a (theta=1)-twisted highest-weight state of the same dimension 4p-2 and charge
-2p+1. We construct 2p W-algebra representations, evaluate their characters,
and show that together with the p-1 integrable representation characters they
generate a modular group representation whose structure is described as a
deformation of the (9p-3)-dimensional representation $R_{p-1} \oplus C^2
\tensor R_{p-1} \oplus R_{p-1} \oplus C^2 \tensor R_{p-1} \oplus C^3 \tensor
R_{p-1}$, where R_{p-1} is the SL(2,Z)-representation on integrable
representation characters and R_{p-1} is a (p+1)-dimensional
SL(2,Z)-representation known from the logarithmic (p,1) model. The dimension
9p-3 is conjecturally the dimension of the space of torus amplitudes, and the
C^n with n=2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra
modules. Under Hamiltonian reduction, the W-algebra currents map into the
currents of the triplet W-algebra of the logarithmic (p,1) model.
