A014090: Numbers that are not the sum of a square and a prime
A014090 ➡️ https://oeis.org/A014090

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A014090.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A014090.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A014090.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

We see the fundamental theorem of arithmetic and the sieve of Eratosthenes in the decomposition into weight × level + jump of natural numbers. Applied to prime numbers this decomposition leads to a new classification of primes.

#decompwlj #math #mathematics #maths #sequence #OEIS #graph #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieveOfEratosthenes #Eratosthenes #sieve #fundamental #theorem #arithmetic #research

A014076: Odd nonprimes
A014076 ➡️ https://oeis.org/A014076

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Odd_nonprimes.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/Odd_nonprimes.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/Odd_nonprimes.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

I've always been wary of "division".

Can I do this? ....

Given \(r \not \equiv 1 \pmod p\) and \(p \not \mid r \) ,

\[ 1 + r + r^2 + \ldots + r^{p-3} + r^{p-2} \equiv \frac{1-r^{p-1}}{1-r} \equiv \frac{1-1}{1-r} \equiv 0 \pmod p \]

This uses Fermat's Little Theorem, \(r^{p-1} \equiv 1 \pmod p\), and noting that the denominator \(1 - r \not \equiv 0 \pmod p\).

The alternative way is to multiply the series on the left by \(1-r\), we know is not congruent to 0, to get \(1-r^{p-1}\) and use FLT to get 0.

#maths #numbertheory

A014011: Defined by a chi-inequality greedy algorithm
A014011 ➡️ https://oeis.org/A014011

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A014011.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A014011.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A014011.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

My first, favorite and most important sequence, the weights of prime numbers: A117078
We see prime numbers classified by level and by weight on the graph.

A117078: a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists ➡️ https://oeis.org/A117078

#decompwlj #math #mathematics #maths #sequence #OEIS #graph #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

A013939: Partial sums of sequence A001221 (number of distinct primes dividing n)
A013939 ➡️ https://oeis.org/A013939

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A013939.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A013939.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A013939.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

If \(a \pmod n\) has order ϕ(𝑛) then 𝑎 is called a primitive root of 𝑛 .

This is the definition in the textbook I'm following, and if differs from what is common on eg wikipedia.

This definition doesn't tell me that a primitive root, if it exists, is always a prime number.

That is, 4 could never be a primitive root but 3 could be.

Is it true? And is the explanation easy?

#maths #numbertheory

A013917: a(n) is prime and sum of all primes <= a(n) is prime
A013917 ➡️ https://oeis.org/A013917

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A013917.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A013917.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A013917.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research

Decomposition into weight × level + jump of prime numbers:
- a new classification of primes ➡️ https://decompwlj.com/primedecomp.php
- in 3D - threejs - webGL ➡️ https://decompwlj.com/3Dgraph/Prime_numbers.html

#decompwlj #math #mathematics #maths #sequence #OEIS #JavaScript #php #graph #3D #threejs #webGL #numbers #primes #PrimeNumbers #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research