11

Consider those natural numbers $n$ such that $7\times10^n+69$ is a prime number. The first $15$ such numbers are $1$, $2$, $3$, $6$, $7$, $8$, $10$, $12$, $13$, $21$, $46$, $68$, $91$, $153$, and $366$. I would say that this is a particulary uninteresting sequence (by the way, this is sequence A294484 in OEIS).

However, it suddenly became interesting to me when I noticed that, for each number $n$ on this sequence, the lights-out problem has a unique solution on a $n\times n$ square board. The numbers for which this property holds form the sequence A076436 in OEIS. The two sequences are not equal; for instance, $15$, $18$, and $20$ belong to the second sequence, but not to the first one. But could it be that every element of the first sequence also belongs to the second one?

José Carlos Santos
  • 443,185
  • Just glancing at it, it seems like that second sequence has quite a few entries, so probably it would 'capture' many other sequences even if there is no direct connection; certainly if we only go up to some point. Indeed, on the face of it, what could possibly make these sequences connected? I am not saying it is impossible, but would you have any idea as to what such a connection might be? What would lights-out have to do with 7 and 69? My money is on coincidence ... but then again, with that attitude no interesting discoveries will ever be made! :) – Bram28 Nov 02 '17 at 15:40
  • My answer to your first question is: I have no idea. However, although I only wrote the first $15$ terms of the first sequence in my question, I checked it up to the $18^\text{th}$ term, and they all belong to the second sequence. If it's a coincidence, it's a fairly interesting one, in my opinion. – José Carlos Santos Nov 02 '17 at 15:44
  • I wonder if anyone is using a computer to try to find connections between all these different sequences ... probably, right? Maybe that'll give us some idea how often these kinds of overlaps occur up to a certain number of entries, between what we, as mathematicians, deem to be interesting sequences. – Bram28 Nov 02 '17 at 15:59
  • A quick and dirty calculation indicates to me that 18 terms coinciding with A076436 by pure coincidence should happen with probability about 1/21000, which isn't large but certainly isn't very small. Because of the difficulty involved in calculating A294484, we may never know the answer to this question. – Ron Kaminsky Apr 19 '18 at 18:17

1 Answers1

0

This is only a partial answer.

Could this be, at least in part, due to different phenomena having a common cause? I notice that, as far as it goes, A076436 contains no $n$ where $n=4\bmod 5$. $n=4\bmod 5\implies 41\mid 7\cdot10^n+69\implies n\not\in A294484$.

Rosie F
  • 3,317
  • On the other hand, elements of most of the other arithmetic series which I found which are excluded from A294484 for similar reasons are found in A076436. E.g., 15 + 16 k, 18 k, 17 + 28 k, 42 + 46 k. – Ron Kaminsky Apr 19 '18 at 19:33
  • Also, thank you for pointing out to me that it is possible that sequences like A294484 are intimately related to a sieving process similar to that which generates the primes. Never thought about that before. – Ron Kaminsky Apr 19 '18 at 20:39