Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

Question

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises:

Write a computer program which

(i) gives each odd number $n<10,000$ that is not a power of a prime and for each prime divisor $p$ of $n$ the corresponding $n_p$ is not forced to be 1 for all groups of order $n$ by the congruence condition of Sylow's Theorem and

(ii) gives for each $n$ in $(i)$ the factorization of $n$ into prime powers and gives the list of all permissible values of $n_p$ for all primes $p$ dividing $n$ (i.e., those values not ruled out by Part 3 of Sylow's Theorem).

and

Carry out the same process as in the preceding exercise for all even numbers less than $1000$. Explain the relative lengths of the lists versus the number of integers tested.

The solutions can be found here and here. It appears that there are $60$ odd candidates in the interval $[1,10,000]$ and $85$ even candidates in the interval $[1,1000]$.

My problem is providing the explanation required for the last part of the second Exercise.

I could find some simple arguments as for why the second list is longer:

• The only even prime powers are powers of $2$, so less numbers are eliminated in the even case.

• For any group $G$ of even order $n$ which is not a power of $2$, Part 3 of Sylow's Theorem leaves all divisors of the index of a Sylow 2-subgroup as possible values for $n_2(G)$. In other words, $n_2$ never eliminates $n$ from the list.

Are these two arguments sufficient to justify the higher density of the even orders? To be more quantitative, The densities are $\frac{60}{10,000}$ for the odd orders vs $\frac{85}{1,000}$ for the even orders. Taking the ratio of the latter by the former makes it seem like the even orders are $\approx 14.2$ times denser.

Can anyone help me give a better explanation as for why the even orders are that much more dense than the odd ones?

Thank you!

Answer 1

Then $n=p_{1}^{\alpha_{1}}\cdots p_{i}^{\alpha_{i}}$ and $n_{p}=kp+1$ such that $n_{p}$ divides $n/p^{\alpha}$ and there is the table of $60$ entries \begin{align*} \begin{array}{ccc} n & p\quad\alpha_{p} & p\quad k\\\hline 105 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 315 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 351 & \left( \begin{array}{cc} 3 & 3 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 13 & 2 \\ \end{array} \right) \\ 495 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 18 \\ 5 & 2 \\ 11 & 4 \\ \end{array} \right) \\ 525 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 58 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 735 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 16 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 945 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 1053 & \left( \begin{array}{cc} 3 & 4 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 13 & 2 \\ \end{array} \right) \\ 1365 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 30 \\ 5 & 4 \\ 5 & 18 \\ 7 & 2 \\ 13 & 8 \\ \end{array} \right) \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 1485 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 1 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 18 \\ 5 & 2 \\ 11 & 4 \\ \end{array} \right) \\ 1575 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 58 \\ 5 & 4 \\ 7 & 2 \\ 7 & 32 \\ \end{array} \right) \\ 1755 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 5 & 70 \\ 13 & 2 \\ \end{array} \right) \\ 1785 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 1 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 28 \\ 3 & 198 \\ 5 & 4 \\ 5 & 10 \\ 7 & 2 \\ 7 & 12 \\ 17 & 2 \\ \end{array} \right) \\ 2025 & \left( \begin{array}{cc} 3 & 4 \\ 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 5 & 16 \\ \end{array} \right) \\ 2205 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 16 \\ 5 & 4 \\ 5 & 88 \\ 7 & 2 \\ \end{array} \right) \\ 2457 & \left( \begin{array}{cc} 3 & 3 \\ 7 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 30 \\ 7 & 50 \\ 13 & 2 \\ \end{array} \right) \\ 2475 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 2 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 18 \\ 5 & 2 \\ 11 & 4 \\ \end{array} \right) \\ 2625 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 3 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 58 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 2775 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 2 \\ 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 12 \\ 3 & 308 \\ 5 & 22 \\ 37 & 2 \\ \end{array} \right) \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 2835 & \left( \begin{array}{cc} 3 & 4 \\ 5 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 5 & 4 \\ 5 & 16 \\ 7 & 2 \\ \end{array} \right) \\ 2907 & \left( \begin{array}{cc} 3 & 2 \\ 17 & 1 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 6 \\ 17 & 10 \\ 19 & 8 \\ \end{array} \right) \\ 3159 & \left( \begin{array}{cc} 3 & 5 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 13 & 2 \\ \end{array} \right) \\ 3393 & \left( \begin{array}{cc} 3 & 2 \\ 13 & 1 \\ 29 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 13 & 20 \\ 29 & 4 \\ \end{array} \right) \\ 3465 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 1 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 18 \\ 3 & 128 \\ 5 & 2 \\ 5 & 4 \\ 5 & 46 \\ 7 & 2 \\ 7 & 14 \\ 11 & 4 \\ \end{array} \right) \\ 3675 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 2 \\ 7 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 16 \\ 3 & 58 \\ 3 & 408 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 3875 & \left( \begin{array}{cc} 5 & 3 \\ 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 6 \\ 31 & 4 \\ \end{array} \right) \\ 4095 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 30 \\ 5 & 4 \\ 5 & 18 \\ 7 & 2 \\ 13 & 8 \\ \end{array} \right) \\ 4125 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 3 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 18 \\ 3 & 458 \\ 5 & 2 \\ 11 & 34 \\ \end{array} \right) \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 4389 & \left( \begin{array}{cc} 3 & 1 \\ 7 & 1 \\ 11 & 1 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 6 \\ 3 & 44 \\ 7 & 8 \\ 11 & 12 \\ 19 & 4 \\ \end{array} \right) \\ 4455 & \left( \begin{array}{cc} 3 & 4 \\ 5 & 1 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 18 \\ 5 & 2 \\ 5 & 16 \\ 5 & 178 \\ 11 & 4 \\ \end{array} \right) \\ 4563 & \left( \begin{array}{cc} 3 & 3 \\ 13 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 3 & 56 \\ 13 & 2 \\ \end{array} \right) \\ 4725 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 2 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 58 \\ 5 & 4 \\ 7 & 2 \\ 7 & 32 \\ \end{array} \right) \\ 4851 & \left( \begin{array}{cc} 3 & 2 \\ 7 & 2 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 16 \\ 7 & 14 \\ 11 & 40 \\ \end{array} \right) \\ 5103 & \left( \begin{array}{cc} 3 & 6 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 104 \\ \end{array} \right) \\ 5145 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 3 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 16 \\ 3 & 114 \\ 5 & 4 \\ 7 & 2 \\ \end{array} \right) \\ 5265 & \left( \begin{array}{cc} 3 & 4 \\ 5 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 5 & 16 \\ 5 & 70 \\ 13 & 2 \\ \end{array} \right) \\ 5313 & \left( \begin{array}{cc} 3 & 1 \\ 7 & 1 \\ 11 & 1 \\ 23 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 84 \\ 3 & 590 \\ 7 & 36 \\ 11 & 2 \\ 23 & 10 \\ \end{array} \right) \\ \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 5355 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 1 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 28 \\ 3 & 198 \\ 5 & 4 \\ 5 & 10 \\ 5 & 214 \\ 7 & 2 \\ 7 & 12 \\ 17 & 2 \\ \end{array} \right) \\ 5445 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 11 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 18 \\ 3 & 40 \\ 5 & 2 \\ 5 & 24 \\ 11 & 4 \\ \end{array} \right) \\ 6075 & \left( \begin{array}{cc} 3 & 5 \\ 5 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 5 & 16 \\ \end{array} \right) \\ 6375 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 3 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 28 \\ 3 & 708 \\ 5 & 10 \\ 17 & 22 \\ \end{array} \right) \\ 6435 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 11 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 3 & 18 \\ 3 & 238 \\ 5 & 2 \\ 11 & 4 \\ 13 & 38 \\ \end{array} \right) \\ 6545 & \left( \begin{array}{cc} 5 & 1 \\ 7 & 1 \\ 11 & 1 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 5 & 2 \\ 7 & 12 \\ 11 & 54 \\ 17 & 2 \\ \end{array} \right) \\ 6615 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 1 \\ 7 & 2 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 16 \\ 5 & 4 \\ 5 & 88 \\ 7 & 2 \\ \end{array} \right) \\ 6669 & \left( \begin{array}{cc} 3 & 3 \\ 13 & 1 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 3 & 6 \\ 3 & 82 \\ 13 & 2 \\ 19 & 2 \\ \end{array} \right) \\ \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 6825 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 2 \\ 7 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 8 \\ 3 & 30 \\ 3 & 58 \\ 3 & 108 \\ 3 & 758 \\ 5 & 4 \\ 5 & 18 \\ 7 & 2 \\ 13 & 8 \\ \end{array} \right) \\ 7371 & \left( \begin{array}{cc} 3 & 4 \\ 7 & 1 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 30 \\ 7 & 50 \\ 13 & 2 \\ \end{array} \right) \\ 7425 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 2 \\ 11 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 18 \\ 5 & 2 \\ 11 & 4 \\ \end{array} \right) \\ 7875 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 3 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 58 \\ 5 & 4 \\ 7 & 2 \\ 7 & 32 \\ \end{array} \right) \\ 8325 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 2 \\ 37 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 8 \\ 3 & 12 \\ 3 & 308 \\ 5 & 22 \\ 37 & 2 \\ \end{array} \right) \\ 8505 & \left( \begin{array}{cc} 3 & 5 \\ 5 & 1 \\ 7 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 5 & 4 \\ 5 & 16 \\ 5 & 340 \\ 7 & 2 \\ \end{array} \right) \\ 8721 & \left( \begin{array}{cc} 3 & 3 \\ 17 & 1 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 6 \\ 17 & 10 \\ 19 & 8 \\ \end{array} \right) \\ 8775 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 2 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 3 & 8 \\ 3 & 108 \\ 5 & 70 \\ 13 & 2 \\ \end{array} \right) \\ 8883 & \left( \begin{array}{cc} 3 & 3 \\ 7 & 1 \\ 47 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 7 & 20 \\ 47 & 4 \\ \end{array} \right) \\ \end{array} \end{align*}

\begin{align*} \begin{array}{ccc} 8925 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 2 \\ 7 & 1 \\ 17 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 8 \\ 3 & 28 \\ 3 & 58 \\ 3 & 198 \\ 5 & 4 \\ 5 & 10 \\ 7 & 2 \\ 7 & 12 \\ 7 & 182 \\ 17 & 2 \\ \end{array} \right) \\ 9045 & \left( \begin{array}{cc} 3 & 3 \\ 5 & 1 \\ 67 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 22 \\ 5 & 40 \\ 67 & 2 \\ \end{array} \right) \\ 9405 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 11 & 1 \\ 19 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 6 \\ 3 & 18 \\ 3 & 348 \\ 5 & 2 \\ 5 & 34 \\ 5 & 376 \\ 11 & 4 \\ 19 & 26 \\ \end{array} \right) \\ 9477 & \left( \begin{array}{cc} 3 & 6 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 4 \\ 13 & 2 \\ 13 & 56 \\ \end{array} \right) \\ 9555 & \left( \begin{array}{cc} 3 & 1 \\ 5 & 1 \\ 7 & 2 \\ 13 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 4 \\ 3 & 16 \\ 3 & 30 \\ 3 & 212 \\ 5 & 4 \\ 5 & 18 \\ 5 & 382 \\ 7 & 2 \\ 13 & 8 \\ \end{array} \right) \\ 9765 & \left( \begin{array}{cc} 3 & 2 \\ 5 & 1 \\ 7 & 1 \\ 31 & 1 \\ \end{array} \right) & \left( \begin{array}{cc} 3 & 2 \\ 3 & 10 \\ 3 & 72 \\ 5 & 4 \\ 5 & 6 \\ 5 & 130 \\ 7 & 2 \\ 7 & 22 \\ 31 & 2 \\ \end{array} \right) \\ \end{array} \end{align*}

The shorter list of evens produces $85$ entries compared to the longer list of odds which produces $60$ entries. The $4$ possible entries of the odds (entries less than $500$) produce $6$ entries on the evens list. Note that many more combinations of odd primes less than $500$ become possible in combination with various powers of $2$. Thus the difference in length of the lists. It is interesting to create a list of the odd prime factorizations less than $500$ and the minimal power of $2$ for which they become solutions such that the overall number is less than $1000$. Let $k$ be the minimal power and $n$ be the the number of powers $k,k+1,\ldots$ that yield solutions less than $1000$. Then there is the table \begin{align*} \begin{array}{cccc} \text{Odd factors} & k & n & \text{Solutions}\\\hline 3 & 2 & 7 & 12,24,48,96,192,384,768\\ 3\cdot 5 & 1 & 6 & 30,60,120,240,480,960\\ 3^{2} & 2 & 5 & 36,72,144,288,576\\ 7 & 3 & 5 & 56,112,224,448,896\\ 5 & 4 & 4 & 80,160,320,640\\ 3^{2}\cdot 5 & 1 & 4 & 90,180,360,720\\ 3^{3} & 2 & 4 & 108,216,432,864\\ 3\cdot 11 & 2 & 3 & 132,264,528\\ 3\cdot 5^{2} & 1 & 3 & 150,300,600\\ 3\cdot 7 & 3 & 3 & 168,336,672\\ 3\cdot 5\cdot 7 & 1 & 3 & 210,420,840\\ 3^{2}\cdot 7 & 2 & 2 & 252,504\\ 3^{3}\cdot 5 & 1 & 2 & 270,540\\ 5\cdot 7 & 3 & 2 & 280,560\\ 3^{2}\cdot 17 & 1 & 2 & 306,612\\ 3^{4} & 2 & 2 & 324,648\\ 5\cdot 19 & 2 & 2 & 380,760\\ 7^{2} & 3 & 2 & 392,784\\ 3^{2}\cdot 11 & 2 & 2 & 396,792\\ 5^{2} & 4 & 2 & 400,800\\ 3^{2}\cdot 5^{2} & 1 & 2 & 450,900\\ 5\cdot 13 & 3 & 1 & 520\\ 3\cdot 7\cdot 13 & 1 & 1 & 546\\ 3\cdot 23 & 3 & 1 & 552\\ 7\cdot 11 & 3 & 1 & 616\\ 3^{2}\cdot 5\cdot 7 & 1 & 1 & 630\\ 3\cdot 5\cdot 11 & 2 & 1 & 660\\ 3^{3}\cdot 13 & 1 & 1 & 702\\ 7\cdot 13 & 3 & 1 & 728\\ 3\cdot 5^{3} & 1 & 1 & 750\\ 3^{3}\cdot 7 & 2 & 1 & 756\\ 3^{4}\cdot 5 & 1 & 1 & 810\\ 3\cdot 11\cdot 13 & 1 & 1 & 858\\ 3\cdot 5\cdot 29 & 1 & 1 & 870\\ 3^{3}\cdot 17 & 1 & 1 & 918\\ 3\cdot 7\cdot 11 & 2 & 1 & 924\\ 3^{5} & 2 & 1 & 972\\ 3^{2}\cdot 5\cdot 11 & 1 & 1 & 990\\ 31 & 5 & 1 & 992\\\hline & & 85 \end{array} \end{align*} Observe that only the odd factorizations $3\cdot 5\cdot 7$, $3^{2}\cdot 5\cdot 7$, $3^{3}\cdot 13$, and $3^{2}\cdot 5\cdot 11$ appeared on the list from Exercise~4.5.47.