Section 13. A person nominated as a candidate for any state, city or town office may withdraw his name from nomination by a request signed and duly acknowledged by him before a notary public, and filed with the officer with whom the nomination was filed, within the time prescribed by section eleven for filing objections to certificates of nomination and nomination papers and no such requests for withdrawals shall be received after such time has expired. A person nominated at a city or town preliminary election may withdraw his name from nomination in the same manner, within six days succeeding five o’clock in the afternoon of the day of holding such preliminary election. Unless otherwise provided by the city or town charter, such nominee shall be replaced by the candidate with the next highest number of votes in said preliminary, if such candidate receives a number of votes at least equal to the number of signatures required by law to place his name on the preliminary election ballot.