Optical frequency synthesizers have widespread applications in optical spectroscopy, frequency metrology, and many other fields. However, their applicability is currently limited by size, cost, and power consumption. Silicon photonics technology, which is compatible with complementary-metal-oxide-semiconductor fabrication processes, provides a low-cost, compact size, lightweight, and low-power-consumption solution. In this work, we demonstrate an optical frequency synthesizer using a fully integrated silicon-based tunable laser. The synthesizer can be self-calibrated by tuning the repetition rate of the internal mode-locked laser. A 20 nm tuning range from 1544 to 1564 nm is achieved with ~10-13 frequency instability at 10 s averaging time. Its flexibility and fast reconfigurability are also demonstrated by fine tuning the synthesizer and generating arbitrary specified patterns over time-frequency coordinates. This work promotes the frequency stability of silicon-based integrated tunable lasers and paves the way toward chip-scale low-cost optical frequency synthesizers.