MathDB
Function with Primes

Source: 2014 AIME II Problem 15

March 27, 2014
functioninductionnumber theoryprime factorizationAMC2014 AIME IIAIME

Problem Statement

For any integer k≥1k\ge1, let p(k)p(k) be the smallest prime which does not divide kk. Define the integer function X(k)X(k) to be the product of all primes less than p(k)p(k) if p(k)>2p(k)>2, and X(k)=1X(k)=1 if p(k)=2p(k)=2. Let {xn}\{x_n\} be the sequence defined by x0=1x_0=1, and xn+1X(xn)=xnp(xn)x_{n+1}X(x_n)=x_np(x_n) for n≥0n\ge0. Find the smallest positive integer, tt such that xt=2090x_t=2090.
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