  We observe that the partition function of the set of all free massless higher
spins s=0,1,2,3,... in flat space is equal to one: the ghost determinants
cancel against the "physical" ones or, equivalently, the (regularized) total
number of degrees of freedom vanishes. This reflects large underlying gauge
symmetry and suggests analogy with supersymmetric or topological theory. The
Z=1 property extends also to the AdS background, i.e. the 1-loop vacuum
partition function of Vasiliev theory is equal to 1 (assuming a particular
regularization of the sum over spins); this was noticed earlier as a
consistency requirement for the vectorial AdS/CFT duality. We find that Z=1 is
also true in the conformal higher spin theory (with higher-derivative d^{2s}
kinetic terms) expanded near flat or conformally flat S^4 background. We also
consider the partition function of free conformal theory of symmetric traceless
rank s tensor field which has 2-derivative kinetic term but only scalar gauge
invariance in flat 4d space. This non-unitary theory has a Weyl-invariant
action in curved background and corresponds to "partially massless" field in
AdS_5. We discuss in detail the special case of s=2 (or "conformal graviton"),
compute the corresponding conformal anomaly coefficients and compare them with
previously found expressions for generic representations of conformal group in
4 dimensions.
