n = 13: 2n+1 = 27 has proper divisors 3 and 9, so we get 93, which has proper divisors 3 and 31, so we get 133.
Then 133 has proper divisors 7 and 19, so we get 917.
Then 917 has proper divisors 7 and 131, so we get 1317.
Then 1317 has proper divisors 3 and 439, so we get 9343, a prime and a(13) = 9343.
Contribution from Sean A. Irvine, Sep 11 2009: (Start)
Proof chain for a(17). The following gives the argument to f at each step, followed by its factorization.
35 factors as 5 * 7.
75 has factors 3 * 5 * 5.
525153 has factors 3 * 193 * 907.
15057112727099753913 has factors 3 * 4463 * 17215189 * 65325353.
179719996575730910515106159846737337176838928854211713151146478934050745192125561494032705883138679506795913535235676554615981512719833136443 has factors 29 * 29 * 5546454298803948416569 * 8370112457804191610629 * 13338101723922940394396774098231 * 345111672681489292530961043464303237918570147336150469919363833
765...4892 (3249 digits) is disible by 2, and hence all subsequent steps will be divisible by 2, therefore no prime is ever reached, therefore a(17)=1. (End)
