Super-d Numbers Revisited

For some reasons, a search for super-\(d\) numbers failed to initially discover a previous post on the topic from February 22nd, 2022. Consequently, some of the content in that post has been repeated. Here is the earlier post titled Super-d Numbers. It's a good idea to view both posts as each contains certain content that isn't repeated in the other. The number of posts in this blog now exceeds 500 so it's easy to forget about previous posts. I need to be thorough in the tags that I add to each post.

Here is the new post created when I wasn't aware of the earlier post. 

For \(d=2, \dots,9\), a super-\(d\) number is a number \(n\) such that \(d \cdot n^d\) contains a substring made of \(d\) digits \(d\). For example, 261 is a super-3 number since \(3\cdot261^3=5\underline{333}8743\).

I was reminded of these numbers because my diurnal age today, 27019, is a super-2 number since \(2 \cdot 27019^2=14600527\underline{22}\). Figure 1 shows a table of the initial \(d\)-numbers for values of \(d\) from 2 to 9.

Figure 1:  source

Figure 2 shows the initial palindromic super-\(d\) numbers for values of \(d\) from 2 to 6.

Figure 2: source

The frequency of super-\(d\) numbers decreases as the value of \(d\) increases. The numbers in the range up to 40,000 are 4377, 420, 43, 12, 1, 0, 0, 0 for \(d\) = 2, 3, 4, 5, 6, 7, 8, 9 respectively. For me, a forthcoming super-6 number, and the only one is the range up to 40,000, is 27257 with the property that:$$6 \cdot 27257 \, ^6=2460478505381 \underline{666666} 506497894 $$Here is a Permalink to the calculation.

Star Numbers Revisited

 So called Star Numbers are few and far between. Up to 40,000, they are:

1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, 11881, 12421, 12973, 13537, 14113, 14701, 15301, 15913, 16537, 17173, 17821, 18481, 19153, 19837, 20533, 21241, 21961, 22693, 23437, 24193, 24961, 25741, 26533, 27337, 28153, 28981, 29821, 30673, 31537, 32413, 33301, 34201, 35113, 36037, 36973, 37921, 38881, 39853

Today I happened to turn 26533 days old which is why my attention was drawn to them again. I had posted about these numbers in an eponymous blog post on 7th June 2019. On that occasion I was writing about the star number 25741 that was 109 days away at that point. I was surprised to see that this post came up in fourth position when the phrase "star numbers" was entered into the Google search bar (see Figure 1):

Figure 1

It also attracted two comments that I only just noticed (see Figure 2). I'm reminded that I should check my various blogs for comments on a regular basis, something that I've quite neglected.

Figure 2

In this post I want to look at some interesting properties of this sequence that I didn't cover in that earlier post. The first involves a result for the sum to infinity of the reciprocals of the star numbers:$$ \sum_{n=1}^{\infty} \frac{1}{S_n}=\frac{\pi \tan{\dfrac{\pi}{2 \sqrt{3}}}}{2 \sqrt{3}} \approx 1.15917331963217$$The second infinite sum involves the factorial function:$$ \sum_{n=0}^{\infty} \frac{S_n}{n!}=7 e$$The third infinite sum involves powers of 2:$$ \sum_{n=1}^{\infty} \frac{S_n}{2^n}=25$$These results are listed in Numbers Aplenty but no proofs are supplied, so I'm just listing them here. Wolfram MathWorld supplies a generating function for the star numbers:

It also provides a linear recurrence relation:  

The star numbers form OEIS :

A003154

Centered 12-gonal (dodecagonal) numbers. Also star numbers: 6*n*(n-1) + 1.

In the comments to this sequence in the OEIS, the diagram shown in Figure 3 appears:

Figure 3

So that's about it. In summary, this post is simply meant to supplement my earlier post and I would encourage anyone reading this post to read that as well, because it contains many other interesting facts about star numbers.