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.