Recamán Sequence

Today I turned 24381* days old and investigation of the number via the Online Encyclopaedia of Integer Sequences (OEIS) brought up reference to the Recamán Sequence that I'd never heard of. The following video does a good job of explaining its significance:

Formally, the Recamán Sequence is defined by OEIS A005132 as:

• a(0) = 0 for n > 0   

• a(n) = a(n-1) - n if positive and not already in the sequence,

• a(n) = a(n-1) + n otherwise

This gives the following initial set of numbers: 

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155

Where 24381 comes into play is that this number marks one of the nth positions in the Recamán sequence for which the ratio A005132(n)/n sets a new record. The OEIS sequence for these record breaking positions is A064622. The eighth member of the Recamán sequence, 20 occurring when n=7, is given as an example: A005132(7)=20, 20/7=2.857, larger than the ratio for any smaller value of n. So 7 is in the sequence.

The initial members of OEIS A064622 are:

1, 2, 3, 6, 7, 19, 34, 67, 102, 115, 190, 2066, 24381, 24398, 24399, 36130540, 409493529, 3744514071

The interesting thing is that position 24381 is closely followed by positions 24398 and 24389 (19 and 20 days respectively from today), after which there is a huge gap until the next record is set.

* as it turned out I didn't turn 24381 days old because I'd been wrongly numbering my days and so 24381 had long passed; however, apart from that the rest of the article is not affected.

Monstrous Moonshine

Via an email alert, I recently came across this question posed in Quora:

Why is 196,884 = 196,883+1 so important?
Can someone explain this in everyday, simple terms?

It was answered by Senia Sheydvasser, P.h.D. Mathematics, Yale University (2017) on Aug 6, 2015 as follows:

Let me sum up the gist of what I wrote here (Senia Sheydvasser's answer to What are some of the most interesting mathematical coincidences?), with emphasis on why this is all so important.

Both 196884 and 196883 were integers that came from important objects in mathematics. 196884 was tied to the j-function, which was important in analytic number theory (speaking roughly: using fancy calculus to answer questions about primes and other integers). 196883, on the other hand, was tied to the Monster group, which was an important object in algebra, specifically in the classification of all finite simple groups. 

Here's the key point: there was no reason to suspect that there was anything at all in common between the j-invariant and the Monster group. They came from completely different fields of study to solve entirely different kinds of problems. And yet... 196884 = 196883 + 1, as John McKay noticed. 

Why were these two integers so close? The initial explanation was that, if you have enough numbers to play around with, you are going to have some coincidences. John McKay was not convinced, and he was right: eventually, people realized that there was deep connection between these two different mathematical fields, which came to be known as monstrous moonshine.

This was the first time I'd heard of the term but it a catchy phrase so I thought I'd investigate further. I came across a video about the topic and I've included it below. The presenter is dreadful but the content involves very high level mathematics and it was way over my head. However, I was familiar with groups and so I thought a good way to approach an understanding would be to first find out more about finite simple groups. Before I go on however, here is the link to the video. 

I choose to revise my understanding of groups by going to Wikipedia, where a wide range of mathematical topics is covered. There is a well-explained and well-illustrated example of a symmetry group that I found helpful. It's a big topic and there's still much to cover but it is still a very active area of mathematical research that impinges on many other disciplines: 

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

I'll keep working my way through this article and hopefully have a better overview of groups at the end of it. Of course, I have numerous text books on the topic that I've collected and to which I can refer to as well if needs be.

Riemann Zeta Function and Analytic Continuation

This is a great video that I came across about the Riemann zeta function and analytic continuation.

WolframAlpha explains analytic continuation as follows:

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is to a complex analytic function determined near a point z_0 by a power series:

Such a power series expansion is in general valid only within its radius of convergence. However, under fortunate circumstances (that are very fortunately also rather common!), the function f will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. This allows, for example, the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic functions from the real line R to the entire complex plane C.

Lucky Numbers

I haven't devoted a entire post to lucky numbers before, even though I have mentioned them (Prime Number Chains). They occur with about the same frequency as prime numbers but whereas WolframAlpha makes a note of prime numbers, it does not mention lucky numbers. So far they just pass by unnoticed. For the sake of completeness, let's define a lucky number once again (taken from WolframAlpha):

Write out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....  

Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/lnN, just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.

The OEIS site also has a list of lucky numbers between 1 and 200000. I've extracted a list of those that are coming up for me:

2495th lucky number: 24727

2496th lucky number: 24729

2497th lucky number: 24733 lucky and prime

2498th lucky number: 24739

2499th lucky number: 24741

2500th lucky number: 24759

2501st lucky number: 24763 lucky and prime

2502nd lucky number: 24771

2503rd lucky number: 24783

2504th lucky number: 24789

2505th lucky number: 24805

2506th lucky number: 24811

2507th lucky number: 24829

2508th lucky number: 24831

2509th lucky number: 24843

2510th lucky number: 24855

2511th lucky number: 24865

2512th lucky number: 24873

2513th lucky number: 24877 lucky and prime

2514th lucky number: 24895

2515th lucky number: 24907 lucky and prime

2516th lucky number: 24909

2517th lucky number: 24933

2518th lucky number: 24951

2519th lucky number: 24957

2520th lucky number: 24963

2521st lucky number: 24985

2522nd lucky number: 24991

I'll begin entering these into my calendar so that I'm reminded on what numbers are lucky as they occur.