Unleashing the Full Potential of SageMath

My new M1 Macbook Air is already proving its usefulness as I discovered when exploring the properties of the number associated with my diurnal age today, namely 27401. This number has a property that qualifies it for membership in OEIS A197816:

A197816

Smallest composite number $m$ such that $m$ and the greatest prime divisor of $m$ begin with $n$.

It took me a while to fully understand what this property involved. Once I did, I developed the algorithm in SageMathCell that is shown in Figure 1 (permalink).

Figure 1

However, the operation times out in SageMathCell which is simply an online implementation of SageMath. In the past, when I used the installation of SageMath on my laptop to address this problem, the laptop would generally freeze up and I would have to reboot it. This laptop was a 2013 Macbook Pro that was clearly not capable of handling the calculations. 

The problem with the algorithm is that after a new value of $m$ is discovered for a given $n$, the value of $n$ needs to reset to 4 every time. This needs to be done 299 times and some of the values for $m$ are quite large. For example, for $n$=114 , the value of $m$ is 114110. Happily my M1 Macbook Air had no difficulty with the calculation and, after 39 seconds, it spat out the numbers for $n$ up to 299. Here is the output:

102, 203, 36, 410, 50, 603, 70, 801, 970, 1010, 110, 1270, 130, 1490, 1510, 1630, 170, 1810, 190, 20030, 2110, 2230, 230, 2410, 2510, 2630, 2710, 2810, 290, 3070, 310, 32030, 3310, 3470, 3530, 3670, 370, 3830, 3970, 4010, 410, 4210, 430, 4430, 4570, 4610, 470, 4870, 4910, 5030, 51010, 5210, 530, 5410, 5570, 5630, 5710, 5870, 590, 6010, 610, 62030, 6310, 6410, 6530, 6610, 670, 6830, 6910, 7010, 710, 7270, 730, 7430, 7510, 7610, 7730, 7870, 790, 8090, 8110, 8210, 830, 84190, 8530, 8630, 8770, 8810, 890, 9070, 9110, 9290, 9370, 9410, 9530, 9670, 970, 9830, 9910, 10090, 1010, 10210, 1030, 10490, 10510, 10610, 1070, 10870, 1090, 11030, 11170, 11230, 1130, 114110, 11510, 11630, 11710, 11810, 11930, 12010, 12130, 12230, 12310, 12490, 12590, 126010, 1270, 12830, 12910, 13010, 1310, 13210, 133090, 134110, 135130, 13610, 1370, 13810, 1390, 14090, 141070, 14230, 14330, 14470, 14510, 146210, 14710, 14810, 1490, 150130, 1510, 15230, 15310, 15430, 15530, 15670, 1570, 15830, 15970, 16010, 16130, 16210, 1630, 164110, 16570, 16630, 1670, 168110, 16930, 17090, 171070, 17210, 1730, 17410, 17530, 176090, 17770, 17830, 1790, 18010, 1810, 18230, 18310, 18470, 185030, 18610, 18710, 18890, 189110, 19010, 1910, 192070, 1930, 19490, 19510, 196030, 1970, 19870, 1990, 20030, 20110, 20270, 20390, 204070, 20530, 20630, 207070, 20810, 20990, 210010, 2110, 21290, 21310, 21410, 21530, 21610, 21790, 218030, 219110, 22030, 22130, 22210, 2230, 22430, 22510, 22670, 2270, 22810, 2290, 23090, 23110, 232010, 2330, 23410, 23510, 236030, 23710, 23810, 2390, 240010, 2410, 24230, 24370, 24410, 24590, 24670, 24730, 248090, 249070, 25030, 2510, 25210, 25310, 25430, 25510, 256010, 2570, 258010, 25910, 26090, 26170, 26210, 2630, 26470, 26570, 26630, 26710, 26830, 2690, 27070, 2710, 27290, 27310, 27410, 27530, 27670, 2770, 27890, 27910, 28010, 2810, 282010, 2830, 28430, 28510, 28610, 28790, 28870, 28970, 29030, 29170, 29270, 2930, 294010, 29530, 29630, 29710, 298030, 29990

Thus 27410 is the first number that begins with 274 and has a greatest prime divisor (2741) that also begins with 274. As the OEIS comments state: a majority of numbers are divisible by 10. SageMathCell is a great online resource and most of the time, for the calculations I carry out, it is sufficient but it's nice to know that for more protracted calculations, the SageMath installation on my laptop can now be relied upon.