pilot light does not stay on on gas burner after vacation
Is the flame hitting the Thermocouple (little pipe to the right of the burner)? This is your safety if it is not hitting that the gas will turn off. These do go bad also and are not very expensive or difficult to change out. After long periods of  not being used I have had to light and hold the pilot then quickly turn on with the controls turned to hot and let the burner run for a few minutes then the system ran OK. This was on a tank less water heater in my shop but similar gas safety (thermocouple). I have also replaced a few TC's only ~$10.