The perturbative expansion of quantum field theory expresses physical quantities as series of numbers (or functions) associated to combinatorial graphs, called Feynman integrals. These integrals are hard to compute, and furthermore their sum forms a series that is in fact divergent. To gain insights into the large order behaviour, Feynman integrals can be approximated astonishingly well by easily computable combinatorial invariants of graphs. I will discuss two such approximations: the tropical Feynman integral and the Martin invariants, using phi^4 theory as an example. The Martin invariants are related to the O(-2) symmetric vector model and can be generalized to an integer sequence. I will end explaining how this sequence encodes the exact value of a Feynman integral through a limit used by Apery to prove the irrationality of zeta(3).