Unsolved problems in number theory | Divisor function | Integer sequences
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple. Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers". A number that is not part of any friendly pair is called solitary. The "abundancy" index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that σ(m) / m = σ(n) / n. "Abundancy" is not the same as abundance, which is defined as σ(n) − 2n. "Abundancy" may also be expressed as where denotes a divisor function with equal to the sum of the k-th powers of the divisors of n. The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy" σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers". In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function. (Wikipedia).
Imaginary numbers are any numbers that include the imaginary number i. A mix of imaginary and real numbers gives you what’s called a complex number. The primary reason we use imaginary numbers is to give us a way to find the root (radical) of a negative number. There’s no way to use real
From playlist Popular Questions
Tutorial - What is an imaginary number
http://www.freemathvideos.com In this video playlist you will learn everything you need to know with complex and imaginary numbers
From playlist Complex Numbers
The World's Favorite Number – Alex Bellos
From playlist G4G11 Videos
MegaFavNumbers: All you need to go Mega is just 3 bytes
Joining the maths #MegaFavNumbers thing just because I like it. My favourity number of over 1 million is a number I remember ever since I was a child. It is used often and well known. Watch to find out why. 16777216
From playlist MegaFavNumbers
We discuss what imaginary numbers are and how they are part of the larger set of complex numbers in this free math video tutorial by Mario's Math Tutoring. This is a nice introduction to working with i. We also go through some examples. 0:26 A Hierarchy of Different Types of Numbers 1:03
From playlist Imaginary & Complex Numbers
Definition of a Critical Number with Examples
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Critical Number with Examples
From playlist Calculus 1 Exam 2 Playlist
Dividing Complex Numbers Example
Dividing Complex Numbers Example Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys
From playlist Complex Numbers
5040 and other Anti-Prime Numbers - Numberphile
Audible: http://www.audible.com/numberphile (free trial) Dr James Grime discusses highly composite numbers. More links & stuff in full description below ↓↓↓ Continues and extra footage: https://youtu.be/PF2GtiApF3E Prime numbers (more videos): http://bit.ly/primevids http://www.antiprim
From playlist Prime Numbers on Numberphile
Matthew Kwan: Friendly bisections of random graphs (CMSA Combinatorics Seminar)
CMSA Combinatorics seminar, 15 September 2021 http://combinatorics-australasia.org/seminars.html
From playlist CMSA Combinatorics Seminar
p-adic Diophantine Approximation with Respect to Fractal Measures by Shreyasi Datta
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
From playlist Ergodic Theory and Dynamical Systems 2022
Oxford MAT 2018 Question 6 (Maths Admissions Test)
These video solutions and more are also now available in this free online course: https://courses.mathsaurus.com/courses/mat 2019 Playlist https://www.youtube.com/playlist?list=PLUuwecyIK3fh6ia74lkh-pPIQWJ38ctIF 2018 Playlist https://www.youtube.com/playlist?list=PLUuwecyIK3fhz00OVD3PhwLG
From playlist Oxford MAT 2018 - Maths Admissions Test
By Their Powers Combined: Sudoku and LSAT // Logic Games [#20] [LSAT Analytical Reasoning]
When I teach LSAT games, one of the ways I introduce them is that they are like sudoku puzzles if you had to build your own grid every time and didn't have enough information to solve the puzzle. So I was pretty delighted when I worked the closest-to-actual-sudoku LSAT game I've ever seen.
From playlist LSAT Games
RedDotRuby 2014 - Fluentd: Data Streams in Ruby World by Satoshi Tagomori
Data streams (ex: logs) are becoming larger, while analytics in (semi-) real time is also becoming more important today. We want to collect huge data sets from many sources, and analyze these data in various way to gain valuable business insights. For these purposes, software on jvm (Hadoo
From playlist RedDotRuby 2014
Top 10 AI Tools for Designers 2023 | Best AI Tools for Designers | AI Tools 2023 | Simplilearn
🔥 Explore Advanced Certification In UI UX Design by Simplilearn: https://www.simplilearn.com/ui-ux-certification-training-course?utm_campaign=26March2023Top10AIToolsforDesigners2023&utm_medium=DescriptionFirstFold&utm_source=youtube 🔥 Explore Artificial Intelligence Engineer (Discount Co
From playlist UI UX Training
25. Structure of set addition V: additive energy and Balog-Szemerédi-Gowers theorem
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Additive energy is a measure of additive structure. Prof.
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Here are the URLs for Genuinely Approachable (GAS) Puzzles 176-180: GAS 176 – Paying Respect by Philip Newman: https://tinyurl.com/y9x45ycd Normal sudoku rules apply. Digits in cages cannot repeat and must sum to the total given. GAS 177 – Thermo Sudoku by Sam: https://tinyurl.com/bnzd2vy
From playlist All the GAS - Genuinely Approachable Sudokus
A LOVELY grammar trick to know how to use -LY ADJECTIVES as adverbs
Adverbs often end in -LY in English but there are a few -LY adjectives too. FREINDLY, LOVELY, LONELY, COWARDLY. So how do we make these into adverbs? There are also 3 adverbs that have 2 forms and -LY and a non-LY form. Late/lately, hard/hardly, near/nearly In this advanced English gramma
From playlist Advanced Grammar tricks - Small changes that make a big difference
My own choice for a number over 1,000,000 is this 617 digit boy: 251959084756578934940271832400483985714292821262040320277771378360436620207075955562640185258807844069182906412495150821892985591491761845028084891200728449926873928072877767359714183472702618963750149718246911650776133798590
From playlist MegaFavNumbers