  Given the widespread use of lossless compression algorithms to approximate
algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression
algorithms fall short at characterizing patterns other than statistical ones
not different to entropy estimations, here we explore an alternative and
complementary approach. We study formal properties of a Levin-inspired measure
$m$ calculated from the output distribution of small Turing machines. We
introduce and justify finite approximations $m_k$ that have been used in some
applications as an alternative to lossless compression algorithms for
approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of
the relevant properties of both $m$ and $m_k$ and compare them to Levin's
Universal Distribution. We provide error estimations of $m_k$ with respect to
$m$. Finally, we present an application to integer sequences from the Online
Encyclopedia of Integer Sequences which suggests that our AP-based measures may
characterize non-statistical patterns, and we report interesting correlations
with textual, function and program description lengths of the said sequences.
