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

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== Problem 24 ==
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== Problem ==
  
 
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 prime factorization of <math>n>1</math>, then <cmath>f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.</cmath>
 
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 prime factorization of <math>n>1</math>, then <cmath>f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.</cmath>
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For every <math>m\ge 2</math>, let <math>f_m(n)=f_1(f_{m-1}(n))</math>. For how many <math>N</math> in the range <math>1\le N\le 400</math> is the sequence <math>(f_1(N),f_2(N),f_3(N),\dots )</math> unbounded?
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For every <math>m\ge 2</math>, let <math>f_m(n)=f_1(f_{m-1}(n))</math>. For how many <math>N</math>s in the range <math>1\le N\le 400</math> is the sequence <math>(f_1(N),f_2(N),f_3(N),\dots )</math> unbounded?
  
 
'''Note:''' A sequence of positive numbers is unbounded if for every integer <math>B</math>, there is a member of the sequence greater than <math>B</math>.
 
'''Note:''' A sequence of positive numbers is unbounded if for every integer <math>B</math>, there is a member of the sequence greater than <math>B</math>.
  
 
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 </math>
 
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 </math>
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== Solution ==
 
== Solution ==
  
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First of all, notice that for any odd prime <math>p</math>, the largest prime that divides <math>p+1</math> is no larger than <math>\frac{p+1}{2}</math>, therefore eventually the factorization of <math>f_k(N)</math> does not contain any prime larger than <math>3</math>. Also, note that <math>f_2(2^m) = f_1(3^{m-1})=2^{2m-4}</math>, when <math>m=4</math> it stays the same but when <math>m\geq 5</math> it grows indefinitely. Therefore any number <math>N</math> that is divisible by <math>2^5</math> or any number <math>N</math> such that <math>f_k(N)</math> is divisible by <math>2^5</math> makes the sequence <math>(f_1(N),f_2(N),f_3(N),\dots )</math> unbounded. There are <math>12</math> multiples of <math>2^5</math> within <math>400</math>. Now let's look at the other cases.
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First of all, notice that for any odd prime <math>p</math>, the largest prime that divides <math>p+1</math> is no larger than <math>\frac{p+1}{2}</math>, therefore eventually the factorization of <math>f_k(N)</math> does not contain any prime larger than <math>3</math>. Also, note that <math>f_2(2^m) = f_1(3^{m-1})=2^{2m-4}</math>, when <math>m=4</math> it stays the same but when <math>m\geq 5</math> it grows indefinitely. Therefore any number <math>N</math> that is divisible by <math>2^5</math> or any number <math>N</math> such that <math>f_k(N)</math> is divisible by <math>2^5</math> makes the sequence <math>(f_1(N),f_2(N),f_3(N),\dots )</math> unbounded. There are <math>12</math> multiples of <math>2^5</math> within <math>400</math>. <math>2^4 5^2=400</math> also works: <math>f_2(2^4 5^2) = f_1(3^4 \cdot 2) = 2^6</math>.
  
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Any first power of prime will not contribute, so at least we look for square of primes. We test primes that are less than <math>\sqrt{400}=20</math>:
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Now let's look at the other cases. Any first power of prime in a prime factorization will not contribute the unboundedness because <math>f_1(p^1)=(p+1)^0=1</math>. At least a square of prime is to contribute. So we test primes that are less than <math>\sqrt{400}=20</math>:
  
 
<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.
  
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<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.
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<math>3^3 \cdot Q^2</math> could also work if <math>Q^2</math> satisfies <math>2~|~f_1(Q^2)</math>, but <math>3^3 \cdot 5^2 > 400</math>.
  
 
<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>.
  
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<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.
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<math>7^2 \cdot Q^2</math> could also work if <math>4~|~f_1(Q^2)</math>, but <math>7^2 \cdot 3^2 > 400</math> already.
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For number that are only divisible by <math>p=11, 13, 17, 19</math>, they don't work because none of these primes are such that <math>p+1</math> could be a multiple of <math>2^5</math> nor a multiple of <math>3^4</math>.
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In conclusion, there are <math>12+1+4+1=18</math> number of <math>N</math>'s ... <math>\framebox{D}</math>.
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==Video Solution by Richard Rusczyk==
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https://artofproblemsolving.com/videos/amc/2012amc12b/276
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~dolphin7
  
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<math>f_1(11^2 \cdot 3) = 2^4 \cdot 3</math> does not work.
 
  
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When prime <math>p\geq 13</math>, any odd multiple <math>p^2</math> other than itself is greater than <math>400</math>, and that <math>f_1(p^2)=p+1</math> could be a multiple of <math>32</math> only if <math>p\geq 31</math>, which is already beyond what we need to test.
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== See Also ==
  
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In conclusion, there are <math>12+4+1=17</math> number of <math>N</math>'s ... <math>\framebox{C}</math>.
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{{AMC12 box|year=2012|ab=B|num-b=23|num-a=25}}
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{{MAA Notice}}

Revision as of 19:57, 10 August 2020

Problem

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded?

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

$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19$

Solution

First of all, notice that for any odd prime $p$, the largest prime that divides $p+1$ is no larger than $\frac{p+1}{2}$, therefore eventually the factorization of $f_k(N)$ does not contain any prime larger than $3$. Also, note that $f_2(2^m) = f_1(3^{m-1})=2^{2m-4}$, when $m=4$ it stays the same but when $m\geq 5$ it grows indefinitely. Therefore any number $N$ that is divisible by $2^5$ or any number $N$ such that $f_k(N)$ is divisible by $2^5$ makes the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded. There are $12$ multiples of $2^5$ within $400$. $2^4 5^2=400$ also works: $f_2(2^4 5^2) = f_1(3^4 \cdot 2) = 2^6$.

Now let's look at the other cases. Any first power of prime in a prime factorization will not contribute the unboundedness because $f_1(p^1)=(p+1)^0=1$. At least a square of prime is to contribute. So we test primes that are less than $\sqrt{400}=20$:

$f_1(3^4)=4^3=2^6$ works, therefore any number $\leq 400$ that are divisible by $3^4$ works: there are $4$ of them.

$3^3 \cdot Q^2$ could also work if $Q^2$ satisfies $2~|~f_1(Q^2)$, but $3^3 \cdot 5^2 > 400$.

$f_1(5^3)=6^2 = 2^2 3^2$ does not work.

$f_1(7^3)=8^2=2^6$ works. There are no other multiples of $7^3$ within $400$.

$7^2 \cdot Q^2$ could also work if $4~|~f_1(Q^2)$, but $7^2 \cdot 3^2 > 400$ already.

For number that are only divisible by $p=11, 13, 17, 19$, they don't work because none of these primes are such that $p+1$ could be a multiple of $2^5$ nor a multiple of $3^4$.

In conclusion, there are $12+1+4+1=18$ number of $N$'s ... $\framebox{D}$.

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012amc12b/276

~dolphin7


See Also

2012 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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