In memory of Fable 5, a deep analysis of the #decompwlj made four days ago with it:
➡️https://decompwlj.com/decompwlj_deep_analysis_report.html
➡️https://web.archive.org/web/20260613101724/https://decompwlj.com/decompwlj_deep_analysis_report.html
#math #AI #Anthropic #Claude #Fable #mathematics #OEIS #sequence #numbers #graph #PrimeNumbers #NumberTheory #sieve #arithmetic #threejs #research #FundamentalTheoremOfArithmetic #sieveOfEratosthenes #classification
A026424: Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative
A026424 ➡️ https://oeis.org/A026424
3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A026424.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A026424.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A026424.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
A026351: a(n) = floor(n*phi) + 1, where phi = (1+sqrt(5))/2
A026351 ➡️ https://oeis.org/A026351
3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A026351.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A026351.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A026351.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
(1) phi(n) is always even for n>=3. ( https://mathworld.wolfram.com/TotientFunction.html )
(2) If n has a primitive root, then it has exactly phi(phi(n)) of them ( https://oeis.org/A010554 )
(3) n has a primitive root if it is of the form 2, 4, p^a, or 2p^a, where p is an odd prime and a>=1 ( https://mathworld.wolfram.com/PrimitiveRoot.html )
From https://oeis.org/A046144, the only numbers with an odd number of primitive roots are 2, 3, 4 and 6.
The totient function phi(n), also called Euler's totient function, is defined as the number of positive integers <=n that are relatively prime to (i.e., do not contain any factor in common with) n, where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function phi(n) can be simply defined as the number of totatives of n. For example, there are eight totatives of 24 (1, 5,...
A025583: Composite numbers that are not the sum of 2 primes
A025583 ➡️ https://oeis.org/A025583
3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A025583.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A025583.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A025583.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 #triangular #numbers #primes #PrimeNumbers #palindromes #animation #FundamentalTheoremOfArithmetic #sequences #NumberTheory #classification #integer #decomposition #number #theory #equation #graphs #sieve #fundamental #theorem #arithmetic #research
A025320: Numbers that are the sum of 2 distinct nonzero squares in 10 or more ways
A025320 ➡️ https://oeis.org/A025320
3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A025320.html
3D graph Gen, threejs animation ➡️ https://decompwlj.com/3DgraphGen/A025320.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A025320.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
Are there always even number of primitive roots?
Because
1. For every primitive root there is a multiplicative inverse that is also a primitive root
2. A primitive root and its inverse are never equivalent
... so we can pair them up.
Does the modulo need to be prime?
I feel like I’ve taken another small step toward understanding primes and the Riemann Hypothesis.
In 1990, Bernard Julia proposed a fascinating physical model called the “Primon Gas.” It treats primes as particles and interprets the Riemann zeta function as the partition function of a thermodynamic system.
In my ongoing “Prime Geography Atlas” project, I’ve been doing large-scale numerical explorations. I discovered two prominent structures at finite scales (organizing cores around log₁₀ ≈ 8.22 and 10.0, connected by a transition layer). I’ve now incorporated these into Julia’s Primon Gas model and formulated them as an Effective Field Theory.
When I consider the **topological stability in the thermodynamic limit**, a path toward the Riemann Hypothesis seems to emerge naturally.
This is still a hypothesis, but it’s an attempt to explain — from a physical perspective — why the beautiful order observed at finite scales might persist all the way to infinity.
If you’re interested, please take a look at the poster.
Feedback and comments from people who love math, physics, or number theory would be greatly appreciated!
#PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #IndependentResearch
Rémi Eismann
恐龍化石 Dino Fossil
Tariq
MoriQ