# All the small primes close together yet again

\begin{align} 2254 & = 2\cdot7\cdot7\cdot23 \\ 2255 & = 5\cdot11\cdot41 \\ 2256 & = 2\cdot2\cdot2\cdot2\cdot3\cdot47 \\ 2257 & = 37\cdot61 \\ 2258 & = 2\cdot1129 \\ 2259 & = 3\cdot3\cdot251 \\ 2260 & = 2\cdot2\cdot5\cdot113 \\ 2261 & = 7\cdot17\cdot19 \\ 2262 & = 2\cdot3\cdot13\cdot29 \\ 2263 & = 31\cdot73 \\ 2264 & = 2\cdot2\cdot2\cdot283 \\ 2265 & = 3\cdot5\cdot151 \\ 2266 & = 2\cdot11\cdot103 \end{align} I this question I pointed something out, of which the foregoing is another instance.

The integer parts of the square roots of all of these numbers are equal to $47$, which is the $15$th prime, so if we're checking for primality, we need to search that far and the rest does itself. Now notice which primes $\le47$ appear above: $$2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\,31,\,37,\,41,\,\bullet,\,47$$ i.e. all of those first $15$ except $43$. Hence nearby numbers can be divisible only by small primes that recur frequently or primes bigger than the square root of the number being factored. (In particular $2272$ is $71$ times a power of $2$, and $2268$ has no prime factors bigger than $7$.)

Is there any reasonable sense it which it could be said that we shouldn't find it so surprising that this---all those early primes occurring so close together---occurs among numbers as small as these? Should we expect frequent instances of this?

PS: "Small" should probably be taken to mean these numbers are not much bigger than $47^2=2209$, where, remember, $47$ is the biggest prime number $\le$ the square roots of these numbers.

As you acknowledge, your example is imperfect in that your set $$S=\{2254,\dots,2266\}$$ lacks a multiple of 43. If we take $$p_i=53$$, then, to cover all primes $$q with a range of consecutive integers not all less than $$p_{i-1}{}^2=47^2=2209$$, we need 16 integers, and the range $$\{2664,\dots,2679\}$$ will do.

To investigate further, for each prime $$p=p_i$$ I found that range of $$k=k(p)$$ consecutive integers $$\{n-k+1,\dots,n\}$$ that covers all primes $$q, minimising $$k(p)$$, subject to $$n\geqslant p_{i-1}{}^2$$.

There are 1229 primes $$p<10000$$. For 258 of them, or about 21%, $$k(p)$$ sets a new record high for $$k$$. Below I list the results for a few of these, where $$k(p)/p$$ also set a record high for $$k/p$$.

$$\begin{array}{rrrrr} p&k&n-k+1&n&k/p\\ 43&14&1763&1776&0.326\\ 103&52&10184&10235&0.505\\ 241&161&56970&57130&0.668\\ 509&358&254472&254829&0.703\\ 1013&783&1023703&1024485&0.773\\ 2113&1807&4456626&4458432&0.855\\ 5101&4476&25996157&26000632&0.877\\ 9241&8441&85369195&85377635&0.913 \end{array}$$

There is a general trend upwards in $$k/p$$. Below is another selection where this time I selected some of those $$p$$ where $$k(p)/p$$ is less than all later $$k(q)/q$$ for all $$q>p$$ within the scope of my program run:

$$\begin{array}{rrrrr} p&k&n-k+1&n&k/p\\ 17&3&285&287&0.176\\ 53&16&2664&2679&0.302\\ 101&37&9699&9735&0.366\\ 229&109&51900&52008&0.476\\ 557&301&307268&307568&0.540\\ 1031&709&1049190&1049898&0.688\\ 2237&1606&4966944&4968549&0.718\\ 5011&4084&25093284&25097367&0.815 \end{array}$$

Your $$p=53$$ is among these! Even though making an $$S$$ which covers all primes $$q<53$$ needed $$k=16$$, somewhat more than your example which lacked 43, $$k(53)=16$$ gives $$k/p\approx0.302$$ which turns out to be a record low.

Whether $$\liminf_{p\to\infty} k(p)/p=1$$ I daren't say, but it appears that way. At any rate, once $$p$$ gets into the thousands, every prime $$p$$ needs a sequence $$S$$ which has (as a proportion) nearly $$p$$ terms.