Using a high-resolution differential technique we have determined the electronic specific heat coefficient gamma(T) of Ba1-xKxFe2As2 with x=0 to 1.0, at temperatures (T) from 2K to 380K and in magnetic fields H=0 to 13T. In the normal state gamma_n(x,T) increases strongly with x at low temperature, compatible with a mass renormalisation ~12 at x=1, and decreases weakly with x at high temperature. A superconducting transition is seen in all samples from x=0.2 to 1, with transition temperatures and condensation energies peaking sharply at x=0.4. Superconducting coherence lengths xi_{ab}~20{AA} and xi_c~3{AA} are estimated from an analysis of Gaussian fluctuations. For many dopings we see features in the H and T-dependences of gamma_s(T,H) in the superconducting state that suggest superconducting gaps in three distinct bands. A broad knee and a sharp mean-field-like peak are typical of two coupled gaps. However, several samples show a shoulder above the sharp peak with an abrupt onset at T_{c,s} and a T-dependence gamma_s(T)proptosqrt{1-T/T_{c,s}}. We provide strong evidence that the shoulder is not due to doping inhomogeneity and suggest it is a distinct gap with an unconventional T-dependence Delta_s(T)propto(1-T/T_{c,s})^{0.75} near T_{c,s}. We estimate band fractions and T=0 gaps from 3-band alpha-model fits to our data and compare the x-dependences of the band fractions with spectroscopic studies of the Fermi surface.
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