Smooth numbers in large prime gaps

<strong>Overview</strong> This dataset contains numbers from 25 up to 1 quadrillion (10¹⁵) that are smooth relative to the gap to the preceding prime. More precisely, we list all numbers <em>n</em> so that <em>r</em> + <em>p^a</em> ≤ <em>n</em> where <em>r</em> is the largest prime smaller than <em>n</em> - 1, and <em>p^a</em> is the largest prime-power divisor of <em>n</em>. The dataset is the result of a 10 day computation using 15 cores on an Intel Xeon system, running code hosted at GitHub (see "Related identifiers"). The GitHub code checks additional conditions when <em>r</em> is <em>n</em> - 2 and <em>n </em>- 1 is a power of 2, but it is easy and quick to check that when (up to 10¹⁵) <em>n</em> = 2^<em>k</em> + 1, the second largest prime <em>r₂</em> satisfies <em>r₂</em> + <em>p^a</em> &gt; n. Thus, this additional check makes no difference in the output. Our motivations for computing this data are described in our paper <em>On invariable generation of alternating groups by elements of prime and prime power order</em> (arXiv:2201.12371). Any number <em>n</em> in the range which is not of the given form has the associated alternating group <em>A_n</em> generated by any element of order <em>r</em> together with any element having a certain cycle structure (and of order <em>p^a</em>). <strong>Description / specification</strong> The data is stored as compressed text-based input to a computer algebra system, specifically in gzipped GAP format. The file out-<em>k</em>.g.gz holds numbers in the range from (<em>k </em>- 1)⋅10¹² to <em>k</em>⋅10¹². The first line of each file sets the variable invgen_oversmooth_range to be the range (thus, [(<em>k </em>- 1)⋅10¹² .. <em>k</em>⋅10¹²]). The subsequent lines set invgen_oversmooth to a list of pairs of numbers [<em>n</em>, <em>p^a</em>], where <em>n</em> is a smooth number as described above, and <em>p^a</em> is the largest prime-power of <em>n</em>. The largest prime preceding <em>n</em> - 1 is given in a GAP comment. Thus, the first few lines of out-0.g.gz (when uncompressed) appear as <pre><code>invgen_oversmooth_range:=[25..1000000000000]; invgen_oversmooth := [ [ 30, 5 ], # bp 23 [ 60, 5 ], # bp 53 [ 126, 9 ], # bp 113 [ 210, 7 ], # bp 199 [ 252, 9 ], # bp 241 [ 308, 11 ], # bp 293 [ 330, 11 ], # bp 317 [ 420, 7 ], # bp 409 ...</code></pre> where [25 .. 1000000000000] is the range considered, and for example "[ 30, 5 ], # bp 23" represents that 23 is the largest prime preceding 30 - 1, 5 is the largest prime-power divisor of 30, and 23 + 5 ≤ 30. We created the data in GAP files for ease of inputting into a GAP program in our own use of the data. It is easy to convert the GAP files to another format via standard technique such as regular expression-based search and replace. For example, on macOS or Linux, the following command will convert the list in out-0.g.gz to a CSV file, which it will display on the terminal. <pre><code class="language-bash">zcat out_quadrillion/out-0.g.gz | sed -En 's/ \[ ([0-9]+), ([0-9]+) \], # bp ([0-9]+)/\1,\2,\3/gp' | less</code></pre>