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%I #4 Mar 30 2012 17:27:32
%S 3,5,31,41,107,11,17,727,499,443,863,439,457,3373,23,1637,53,6857,31,
%T 47,5323,811,6911,919,29,19681,439,739,13499,29789,43,7187,43,461,
%U 23327,50651,59,2579,2909,22973,2179,15901,14197,293,1187,34607,11059
%N Smallest prime p such that x = n is a solution mod p of x^3 = 2, or 0 if no such prime exists.
%C Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^3 = 2 iff n^3-2 has a prime factor > n; n is a solution mod p of x^3 = 2 iff p is a prime factor of n^3-2 and p > n.
%C n^3-2 has at most two prime factors > n, consequently these factors are the only primes p such that n is a solution mod p of x^3 = 2. For n such that n^3-2 has no prime factor > n (the zeros in the sequence; they occur beyond the last entry shown in the database) see A060591. For n such that n^3-2 has two prime factors > n, cf. A060914.
%F If n^3-2 has prime factors > n, then a(n) = least of these prime factors, else a(n) = 0.
%e a(2) = 3, since 2 is a solution mod 3 of x^3 = 2 and 2 is not a solution mod p of x^3 = 2 for prime p = 2. Although 2^3 = 2 mod 2, prime 2 is excluded because 0 < 2 and 2 = 0 mod 2. a(5) = 41, since 5 is a solution mod 41 of x^3 = 2 and 5 is not a solution mod p of x^3 = 2 for primes p < 41. Although 5^3 = 2 mod 3, prime 3 is excluded because 3 < 5 and 5 = 2 mod 3.
%Y Cf. A040028, A060121, A060122, A060123, A060124, A060591, A060914.
%K nonn
%O 2,1
%A _Klaus Brockhaus_, Mar 02 2001