The Quest for 700: Weekly GMAT Challenge

The function p(n) on non-negative integer n is defined in the following way: the units digit of n is the exponent of 2 in the prime factorization of p(n), the tens digit is the exponent of 3, and in general, for positive integer k, the digit in the 10^k–1th place of n is the exponent on the kth smallest prime (compared to the set of all primes) in the prime factorization of p(n). For instance, p(102) = 20, since 20 = 5¹3⁰2². What is the smallest positive integer that is not equal to p(n) for any permissible n?

(A) 1
(B) 29

(C) 31

(D) 1,024

(E) 2,310