# Difference between revisions of "2012 AMC 12B Problems/Problem 24"

(Created page with "== Problem 24 == Define the function <math>f_1</math> on the positive integers by setting <math>f_1(1)=1</math> and if <math>n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}</math> is the p...") |
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<math>f_1(3^4)=4^3=2^6</math> works, therefore any number <math>\leq 400</math> that are divisible by <math>3^4</math> works: there are <math>4</math> of them. | <math>f_1(3^4)=4^3=2^6</math> works, therefore any number <math>\leq 400</math> that are divisible by <math>3^4</math> works: there are <math>4</math> of them. | ||

− | <math> | + | <math>3^3 \cdot Q^2</math> could also work if <math>Q</math> is odd, but <math>3^3 \cdot 5^2 >400</math> already. |

<math>f_1(5^3)=6^2 = 2^2 3^2</math> does not work. | <math>f_1(5^3)=6^2 = 2^2 3^2</math> does not work. | ||

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<math>f_1(7^3)=8^2=2^6</math> works. There are no other multiples of <math>7^3</math> within <math>400</math>. | <math>f_1(7^3)=8^2=2^6</math> works. There are no other multiples of <math>7^3</math> within <math>400</math>. | ||

− | <math> | + | <math>7^2 \cdot Q^3</math> could also work if <math>Q</math> is odd, but <math>7^2 \cdot 3^3 > 400</math> already. |

<math>f_1(11^2 \cdot 3) = 2^4 \cdot 3</math> does not work. | <math>f_1(11^2 \cdot 3) = 2^4 \cdot 3</math> does not work. |

## Revision as of 04:16, 6 December 2012

## Problem 24

Define the function on the positive integers by setting and if is the prime factorization of , then For every , let . For how many in the range is the sequence unbounded?

**Note:** A sequence of positive numbers is unbounded if for every integer , there is a member of the sequence greater than .

## Solution

First of all, notice that for any odd prime , the largest prime that divides is no larger than , therefore eventually the factorization of does not contain any prime larger than . Also, note that , when it stays the same but when it grows indefinitely. Therefore any number that is divisible by or any number such that is divisible by makes the sequence unbounded.

Numbers that are divisible by : .

Any first power of prime will not contribute, so at least we look for square of primes. We test primes that are less than :

works, therefore any number that are divisible by works: there are of them.

could also work if is odd, but already.

does not work.

works. There are no other multiples of within .

could also work if is odd, but already.

does not work.

When prime , any odd multiple other than itself is greater than , and that could be a multiple of only if , which is already beyond what we need to test.

In conclusion, there are number of 's ... .