OEIS Sequences

What is OEIS?

OEIS, or The Online Encyclopedia of Integer Sequences, is an online database of numerous number sequences found to date. Currently, the database contains approximately 326,000 sequences, and is constantly being added to and expanded upon by users finding new sequences and developing richer explanations to older ones.

OEIS was originally created by Neil James Alexander Sloane, who, at the time he had started collecting and organizing sequences, was a researcher at AT&T Labs. While the database today is maintained by the OEIS Foundation (est. 2009), Sloane remains the president of the foundation to today and continues to organize sequences into the database and aid in expanding the archive.

While the database contains far more sequences, all with their own unique behaviours and interesting mathematical formula(s) to be represented by, here are a few of which have appealed to me while browsing through it:

OEIS Sequence A048724

A048724 is a sequence that revolves around the XOR operator being applied to the current x value n and 2n. It was discovered by Antti Karttunen in April 26, 1999, and is currently labeled under the keywords nonn, nice, look, and easy on OEIS. The formula for A048724 is as follows:

$$a(n) = n \underline{\vee} 2n$$

OEIS Sequence A006577

With a cool pattern and a relatively simple formula, A006577 is a sequence based on the Collatz Conjecture, which states that every positive integer is capable of reaching 1 by simply dividing itself by 2 if it is even and multiplying itself by 3 and adding 1 if it is odd. This conjecture is also famously known as the $3n+1$ problem. While it has no explicit mathematical formula, A006577 is a sequence where a(n) is equal to the number of times needed to reach 1 starting from n following the Collatz Conjecture.

For example, a(13) would equal 9 considering the sequence would follow these steps:

13 * 3 + 1 = 40
40 / 2 = 20
20 / 2 = 10
10 / 2 = 5
5 * 3 + 1 = 16
16 / 2 = 8
8 / 2 = 4
4 / 2 = 2
2 / 2 = 1

The sequence was originally published by N.J.A. Sloane on July 11, 1991, and is currently categorized under the keywords nonn, nice, easy, hear, and look on OEIS.

OEIS Sequence A030101

A030101 as a graph is a fairly odd visual; how are numbers capable of being organized in such a way that they create numerous rectangles where the sizes gradually increase each time a new one is created? Surprisingly, however, this sequence is quite simple, and is simply the reversed binary value of any value of n within the graph, where n is a positive integer greater than 0.

For example, a(13) would be equal to 11 because the binary representation of 13 is 1101, which reversed is equal to 1011 or 11 in base 10.

A030101 was originally added to the database by David W. Wilson in 1999, and has since been categorized under the keywords nonn, base, nice, hear, and look in the OEIS.

OEIS Sequence A002618

A002618 is a sequence that revolves around Euler's phi function. In terms of number theory, $\phi(n)$ is a mathematical function that returns the number of integers less than n and greater than 1 that are relatively prime to n. Using this function, we can create a concise formula that represents the sequence graphed above:

$$a(n) = n * \phi(n)$$

This formula appears to graph numerous scattered points which seem to form several individual exponential functions, all diverging from a single point.

A002618 was originally published on OEIS on April 30, 1991 by N.J.A. Sloane, and has since been categorized under the keywords nonn, easy, nice, mult, and look.

OEIS Sequence A005185

A005185 is a sequence that is based heavily on recursion, and appears to graph an ever expanding sequence of diagonally-facing rectangles. The sequence itself follows a simple, yet somewhat confusing formula, where it is assumed that $a(1) = a(2) = 1$ and $a > 2$:

$$a(n) = a(n-a(n-1)) + a(n-a(n-2))$$

A005185 was originally added to the OEIS on May 20, 1991 by Simon Plouffe and N.J.A. Sloane, and is currently categorized under the keywords nonn, nice, easy, and look.

Faisal Alsaidi 2019