TITLE: $1$ as difference of composites with same number of prime factors and smallest examples
QUESTION [12 upvotes]: It is probably open can we for every $k \in \mathbb N$ find two composites $a_k$ and $b_k$ such that $a_k$ and $b_k$ have exactly $k$ prime factors and $a_k-b_k=1$.
Smallest examples found so far are:
for $k=1$ $$3^2-2^3=1$$ for $k=2$ $$3 \cdot 5 - 7 \cdot 2=1$$ for $k=3$ $$3\cdot7\cdot11-2\cdot5\cdot23=1$$ for $k=4$ $$5 \cdot 7 \cdot 11 \cdot 19 - 2 \cdot 3 \cdot 23 \cdot 53=1$$ for $k=5$ $$3\cdot7\cdot17\cdot23\cdot31-2^2\cdot5\cdot11\cdot13\cdot89=1$$
It is easy to see that $3$ is a factor of one of numbers that are smallest pair for $k=1,2,3,4,5$ so I will make a pretty dumb conjecture that could be ruled out with a clever computer-check:

If a pair $(a_k,b_k)$ is smallest pair then $3$ is a factor of one of those two numbers.

Until which $k$ is this true?
I am prepared to let this always be true because smallest pairs should tend to have small primes as factors of them and because there is a good chance that one of members is divisible by $3$ because they differ only by one.
I am not sure would I like to see a counterexample, but go for it.
This is true at least for $k=1,2,3,4,5,6,7,8,9,10,11$ by this list .

REPLY [8 votes]: I'll write currently the smallest known for me pairs $(a_k,b_k)$ 
for $k=12,...,19$ (so far).
 Hope it will help to optimize search of the smallest examples.
(underlined parts of $a_k$, $b_k$ prime factorizations are constructed 
of first prime numbers)
$k= 12$:
 $a_k = 12\:633\:565\:576\:364\:596\:150$;
 $\log_2(a_k)\approx 63.4539$;
 $a_k = \underline{2 \cdot 5^2 \cdot 7 \cdot 17 \cdot 29} 
  \cdot 41 \cdot 47  \cdot 79  \cdot 97  \cdot 107  \cdot 149  \cdot 311$;
 $b_k = \underline{3 \cdot 11  \cdot 13  \cdot 19  \cdot 23} 
  \cdot 43 \cdot 59  \cdot 61  \cdot 71  \cdot 89  \cdot 137  \cdot 503$;
$ $
$k=13$ (solution by Peter Košinár):
 $a_k = 2\:909\:526\:601\:415\:962\:299\:135$;
 $\log_2(a_k) \approx 71.3013$;
 $a_k = \underline{3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 31} \cdot 61 
  \cdot 73 \cdot 97 \cdot 199 \cdot 271 \cdot 401 \cdot 433$; 
 $b_k = \underline{2 \cdot 11 \cdot 19 \cdot 23 \cdot 29 \cdot 37 \cdot 41}
  \cdot 59 \cdot 67 \cdot 83 \cdot 167 \cdot 223 \cdot 563$;
$ $
$k=14$:
 $a_k = 561\:562\:786\:476\:519\:254\:711\:571$;
 $\log_2(a_k) \approx 78.8938$;
 $a_k = \underline{3 \cdot 7 \cdot 17 \cdot 23} \cdot 37 \cdot 41 
  \cdot 53 \cdot 59 \cdot 61 \cdot 89 \cdot 101 \cdot 137 \cdot 281 \cdot 683$;
 $b_k = \underline{2 \cdot 5 \cdot 11^2 \cdot 13 \cdot 19 \cdot 29} 
  \cdot 43 \cdot 47 \cdot 83 \cdot 97 \cdot 173 \cdot 193 \cdot 239 \cdot 499$;
$ $
$k=15$:
 $a_k = 85\:227\:871\:774\:666\:834\:526\:775\:385$;
 $\log_2(a_k) \approx 86.1395$;
 $a_k = \underline{5 \cdot 7 \cdot 11 \cdot 23 \cdot 31 \cdot 37 \cdot 41}
  \cdot 61 \cdot 71 \cdot 109 \cdot 137 \cdot 139 \cdot 179 \cdot 193 \cdot 659$;
 $b_k = \underline{2^3 \cdot 3 \cdot 13 \cdot 17 \cdot 19 \cdot 29} \cdot 47
  \cdot 67 \cdot 79 \cdot 97 \cdot 113 \cdot 227 \cdot 263 \cdot 359 \cdot 499$;
further pairs look not as candidates to be 'the smallest'; 
they are written just as current upper bound:
$k=16$:
 $a_k = 47\:378\:556\:835\:179\:047\:024\:124\:411\:471$;
 $\log_2(a_k)  \approx 95.2582$;
 $a_k = \underline{3 \cdot 11 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 53 
  \cdot 67 \cdot 71 \cdot 73 \cdot 79} \cdot 157 \cdot 229 \cdot 251 \cdot 337 
  \cdot 383$;
 $b_k = \underline{2 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 19 \cdot 37 
  \cdot 43 \cdot 47 \cdot 59 \cdot 61} \cdot 109 \cdot 199 \cdot 509 
  \cdot 2131 \cdot 25457$;
$ $
$k=17$:
 $a_k = 35\:516\:659\:231\:848\:431\:785\:222\:277\:597\:395$;
 $\log_2(a_k)  \approx 104.808$;
 $a_k = \underline{5 \cdot 17 \cdot 19 \cdot 43 \cdot 47 \cdot 53 \cdot 59
  \cdot 61} \cdot 71 \cdot 79 \cdot 83 \cdot 103 \cdot 107 \cdot 149 \cdot 151
  \cdot 331 \cdot 1493$;
 $b_k = \underline{2 \cdot 3^3 \cdot 7 \cdot 11 \cdot 13 \cdot 23 \cdot 29
  \cdot 31 \cdot 37 \cdot 41} \cdot 89 \cdot 97 \cdot 409 \cdot 541 \cdot 761
  \cdot 2053 \cdot 7019$;
$ $
$k=18$:
 $a_k = 4\:125\:576\:774\:669\:197\:249\:512\:925\:965\:978\:465$;
 $\log_2(a_k)  \approx 111.668$;
 $a_k = \underline{5 \cdot 11 \cdot 23 \cdot 29 \cdot 41 \cdot 43 \cdot 47
  \cdot 59 \cdot 67 \cdot 71 \cdot 73 \cdot 83 \cdot 97} \cdot 139 \cdot 211
  \cdot 431 \cdot 683 \cdot 953$;
 $b_k = \underline{2^5 \cdot 3 \cdot 7 \cdot 13 \cdot 17 \cdot 19 \cdot 31
  \cdot 37 \cdot 53 \cdot 61 \cdot 79 \cdot 89} \cdot 137 \cdot 163 \cdot 349
  \cdot 983 \cdot 2617 \cdot 2797$;
$ $
$k=19$:
 $a_k = 13\:656\:266\:430\:033\:430\:075\:461\:309\:350\:278\:726\:155$;
 $\log_2(a_k)  \approx 123.361$;
 $a_k = \small{\underline{3 \cdot 5 \cdot 13 \cdot 17 \cdot 23 \cdot 31
  \cdot 47 \cdot 53 \cdot 59 \cdot 101 \cdot 103 \cdot 109 \cdot 113}
  \cdot 127 \cdot 241 \cdot 307 \cdot 659 \cdot 2131 \cdot 23251}$;
 $b_k = \small{\underline{2 \cdot 7^2 \cdot 11^2 \cdot 19 \cdot 29
  \cdot 37 \cdot 41 \cdot 43 \cdot 61 \cdot 71 \cdot 79 \cdot 83 \cdot 89
  \cdot 97} \cdot 191 \cdot 257 \cdot 709 \cdot 1153 \cdot 3257}$.

We can define theoretical (unreachable) lower bound for given $k$ as
$$m_k = \sqrt{p_{2k}\#},$$
where $p_n\#$ is primorial ($p_n\# = p_1\cdot p_2\cdots p_n$).
This way we can estimate efficiency of incoming new values:
 denote 'defect' $\Delta$: $$\Delta = \log_2(a_k)-\log_2(m_k)$$ 
(if 'defect' is less, then estimation is better); 
 one can note that $\Delta$ growth more-or-less linearly in the range 
$k=2 \ldots 15$, so the values for $k=12,13,14,15$ are not far 
from the smallest examples:
\begin{array}{|l|r|l|l|}
\hline
k & \log_2(m_k) & \log_2(a_k) & \Delta \\
\hline
... & ... & ... & ... \\
7  & 26.7693 & 29.8460 & 3.077\\
8  & 32.4105 & 35.9390 & 3.528 \\
9  & 38.3172 & 42.3748 & 4.058 \\
10 & 44.4251 & 48.7998 & 4.375 \\
11 & 50.6719 & 56.0886 & 5.417 \\
\hline
12 & 57.0973 & 63.4539 & 6.357 \\
13 & 63.7263 & 71.3013 & 7.575 \\
14 & 70.4404 & 78.8938 & 8.453 \\
15 & 77.2345 & 86.1395 & 8.905 \\
16 & 84.2455 & 95.2582 & 11.013 \\
17 & 91.3541 & 104.808 & 13.454 \\
18 & 98.5829 & 111.668 & 13.085 \\
19 & 105.905 & 123.361 & 17.456 \\ 
... & ... & ... & ... \\
\hline
\end{array} 
And plot of the current values $\Delta$: