Skip to main content
MathOverflow

Return to Answer

edited body
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $13 \leq p \leq 19$$13 \leq p \leq 23$ prime and $q\in\mathbb{N}^+$ odd

  • $3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $17 \leq p \leq 23$$17 \leq p \leq 19$ prime and $q\in\mathbb{N}^+$ odd

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $13 \leq p \leq 19$ prime and $q\in\mathbb{N}^+$ odd

  • $3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $17 \leq p \leq 23$ prime and $q\in\mathbb{N}^+$ odd

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $13 \leq p \leq 23$ prime and $q\in\mathbb{N}^+$ odd

  • $3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $17 \leq p \leq 19$ prime and $q\in\mathbb{N}^+$ odd

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$
deleted 139 characters in body
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 5\cdot p$ with $5 < p \leq 13$

  • $3^{k_0} \cdot 5^{k_1} \cdot p^{k_2} \cdot q$ with $q$ odd, $k_i \geq 1$

    $3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 7\cdot 11 \cdot p$$3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $11 < p \leq 19$

  • $3^{k_0} \cdot 7^{k_1} \cdot 11^{k_{2}} \cdot p^{k_{3}} \cdot q$ with$13 \leq p \leq 19$ prime and $q$$q\in\mathbb{N}^+$ odd, $k_i \geq 1$

  • $3\cdot 7\cdot 13 \cdot p$$3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $13 < p \leq 23$

  • $3^{k_0} \cdot 7^{k_1} \cdot 13^{k_{2}} \cdot p^{k_{3}} \cdot q$ with$17 \leq p \leq 23$ prime and $q$$q\in\mathbb{N}^+$ odd, $k_i \geq 1$

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 5\cdot p$ with $5 < p \leq 13$

  • $3^{k_0} \cdot 5^{k_1} \cdot p^{k_2} \cdot q$ with $q$ odd, $k_i \geq 1$

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 7\cdot 11 \cdot p$ with $11 < p \leq 19$

  • $3^{k_0} \cdot 7^{k_1} \cdot 11^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

  • $3\cdot 7\cdot 13 \cdot p$ with $13 < p \leq 23$

  • $3^{k_0} \cdot 7^{k_1} \cdot 13^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 5 \cdot p \cdot q$ with $7 \leq p \leq 13$ prime and $q\in\mathbb{N}^+$ odd

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3 \cdot 7 \cdot 11 \cdot p \cdot q$ with $13 \leq p \leq 19$ prime and $q\in\mathbb{N}^+$ odd

  • $3 \cdot 7 \cdot 13 \cdot p \cdot q$ with $17 \leq p \leq 23$ prime and $q\in\mathbb{N}^+$ odd

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$
added 371 characters in body
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 5\cdot p$ with $5 < p \leq 13$

  • $3^{k_0} \cdot 5^{k_1} \cdot p^{k_2} \cdot q$ with $q$ odd, $k_i \geq 1$

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 7\cdot 11 \cdot p$ with $11 < p \leq 19$

  • $3^{k_0} \cdot 7^{k_1} \cdot 11^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

  • $3\cdot 7\cdot 13 \cdot p$ with $13 < p \leq 23$

  • $3^{k_0} \cdot 7^{k_1} \cdot 13^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

 

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 5\cdot p$ with $5 < p \leq 13$

  • $3^{k_0} \cdot 5^{k_1} \cdot p^{k_2} \cdot q$ with $q$ odd, $k_i \geq 1$

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 7\cdot 11 \cdot p$ with $11 < p \leq 19$

  • $3^{k_0} \cdot 7^{k_1} \cdot 11^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

  • $3\cdot 7\cdot 13 \cdot p$ with $13 < p \leq 23$

  • $3^{k_0} \cdot 7^{k_1} \cdot 13^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

I'd like to sum up some specific parts of Wojowu's answer:

  1. The smallest odd $n$ with $\varphi(n)/n = \prod_{p|n}(1-1/p) \leq 1/2$ is $105 = 3\cdot 5\cdot 7$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 5\cdot p$ with $5 < p \leq 13$

  • $3^{k_0} \cdot 5^{k_1} \cdot p^{k_2} \cdot q$ with $q$ odd, $k_i \geq 1$

  1. The smallest odd $n$ with $\varphi(n)/n \leq 1/2$ not found yet is $3003 = 3\cdot 7 \cdot 11 \cdot 13$

  2. $\varphi(n)/n < 1/2$ also for

  • $3\cdot 7\cdot 11 \cdot p$ with $11 < p \leq 19$

  • $3^{k_0} \cdot 7^{k_1} \cdot 11^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

  • $3\cdot 7\cdot 13 \cdot p$ with $13 < p \leq 23$

  • $3^{k_0} \cdot 7^{k_1} \cdot 13^{k_{2}} \cdot p^{k_{3}} \cdot q$ with $q$ odd, $k_i \geq 1$

 

The natural question then is: What comes next, what is the smallest $n$ with $\varphi(n)/n \leq 1/2$ not found yet? And so on, and so on. (I have to admit, @Wojowu, that I'm still not able to answer this.)

Some shortest sequences of primes $p_1,\dots,p_k$ with $\prod_{p_i}(1-1/p_i) \leq 1/2$ and their products:

  • $3\cdot 11 \cdot 13 \cdot 17 \cdot 19 = 138,567$ (is this the next one?)
  • $3 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 = 260,468,169$
  • $5\cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 = 37,182,145$
added 95 characters in body
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113
added 4 characters in body
Source Link
Greg Martin
Greg Martin
  • 12.8k
  • 1
  • 48
  • 72
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113