Arithmetic Derivative Records

On June 19th 2019 I made a post titled Arithmetic Derivative and in this post I'll return to the topic. I was prompted to do so by one of the properties of the number associated with my diurnal age today. The number is 27265 and it has the property that its arithmetic derivative, 11448, has no digits in common with the number itself. I've listed these numbers in an entry in my Bespoken for Sequences.

I'm not going to pursue that topic in this post but I was reminded of arithmetic derivatives and got to thinking about records being set by the size of arithmetic derivatives as the natural numbers are traversed. It didn't take long to develop a SageMath algorithm to explore this topic (permalink). Detailed results are shown below in Table 1 with derivatives up 40000 in size, although the algorithm lists the records for numbers up to one million.

Table 1

The record setting sizes of the arithmetic derivatives form OEIS A131116 and the initial values are as follows:

4, 5, 12, 16, 32, 44, 80, 112, 192, 272, 448, 640, 1024, 1472, 2304, 2368, 3328, 3392, 5120, 5376, 7424, 7744, 11264, 12032, 16384, 16640, 17408, 24576, 26624, 35840, 36864, 38656, 53248, 58368, 77824, 80896, 84992, 114688, 126976, 167936, 176128, 185344, 245760, 274432, 360448, 380928, 401408, 524288, 528384, 589824, 593920, 770048, 774144, 819200, 864256, 1114112, 1130496, 1261568, 1277952, 1638400, 1658880, 1753088, 1851392, 2359296, 2408448, 2686976, 2736128, 3473408, 3538944, 3735552, 3948544, 4980736, 5111808, 5701632, 5832704, 7340032, 7520256, 7929856, 8388608

Table 2 shows a graph of these numbers:

Table 2

As can be seen from Table 1, all the numbers associated with these records contain all powers of 2 together with multiples of these powers. The initial numbers are:

4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3584, 3840, 4096, 5120, 6144, 7168, 7680, 8192, 10240, 12288, 14336, 15360, 16384, 20480, 24576, 28672, 30720, 32768, 40960, 49152, 57344, 61440, 65536, 73728, 81920, 90112, 98304, 110592, 114688, 122880, 131072, 147456, 163840, 180224, 196608, 221184, 229376, 245760, 262144, 294912, 327680, 360448, 393216, 442368, 458752, 491520, 524288, 589824, 655360, 720896, 786432, 884736, 917504, 983040

Taylor Series

For a really superb video explaining how Taylor Polynomials and Taylor Series are created, you need go no further than this video by 3Blue1Brown.

$$ \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} =f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+ \cdots $$The formula above was copied from Wikipedia where all mathematical expressions are in LaTeX but enclosed within <math> ... </math> tags. While it's always good practice to create the LaTeX from scratch, it's still useful to remember that any mathematical expressions found in Wikipedia can be copied directly into Blogger.

Anyway, the main point of this post is to remind myself to watch more of 3Blue1Brown's videos. The Taylor Series video was created in May of 2017 and since then twenty more have been added. Altogether he has almost seventy videos of which I've watched sixteen. The description of the channel states that:

3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective. 

He can be followed on Twitter at https://twitter.com/3blue1brown and a website at http://www.3blue1brown.com.