One day, one decomposition
A242414: Numbers whose prime factorization viewed as a tuple of nonzero powers is palindromic

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A242414.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A242414.html

One day, one decomposition
A212168: Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n))

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A212168.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A212168.html

One day, one decomposition
A212166: Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k))

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A212166.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A212166.html

One day, one decomposition
A212165: Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k))

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A212165.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A212165.html

One day, one decomposition
A212164: Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k))

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A212164.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A212164.html

'Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery', by Zhen Qin, Michael B. Wakin, Zhihui Zhu.

http://jmlr.org/papers/v25/24-0029.html

#tensor #tensors #factorization

Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/

$\text{Let}{p}_n=\text{the}{n}^{\text{th}}\text{prime}.$

$\text{Then}2025=81\cdot 25=3^4{5}^2$

$={p_2}^4{p_3}^2={p_2}^{{p_1}^{p_1}}{p_3}^{p_1}={p_{p_1}}^{{p_1}^{p_1}}{p_{p_2}}^{p_1}={p_{p_1}}^{{p_1}^{p_1}}{p_{p_{p_1}}}^{p_1}$

No information is lost by dropping the terminal 1s. Thus we may write the following form.

$$2025={p_p}^{p^p}{p_{p_p}}^p$$

The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.

Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png

Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png

Reference —

Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

'Triple Component Matrix Factorization: Untangling Global, Local, and Noisy Components', by Naichen Shi, Salar Fattahi, Raed Al Kontar.

http://jmlr.org/papers/v25/24-0400.html

#minimization #factorization #sparse

One day, one decomposition
A130091: Numbers having in their canonical prime factorization mutually distinct exponents

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A130091.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A130091.html

I previously helped discover that RSA keys in certificates generated by printers from Canon and Fuji could be factorized with the centuries-old Fermat factorization algorithm https://fermatattack.secvuln.info/ #rsa #factorization #fermat

One day, one decomposition
A073493: Numbers having exactly one prime gap in their factorization

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A073493.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A073493.html

One day, one decomposition
A073492: Numbers having at least one prime gap in their factorization

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A073492.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A073492.html

'Scale Invariant Power Iteration', by Cheolmin Kim, Youngseok Kim, Diego Klabjan.

http://jmlr.org/papers/v24/19-784.html

#factorization #optimization #algorithms

One day, one decomposition
A046760: Wasteful numbers

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/Wasteful_numbers.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/Wasteful_numbers.html

One day, one decomposition
A046759: Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)<l(n)

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/Economical_numbers.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/Economical_numbers.html

One day, one decomposition
A046758: Equidigital numbers

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/Equidigital_numbers.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/Equidigital_numbers.html

'Multi-source Learning via Completion of Block-wise Overlapping Noisy Matrices', by Doudou Zhou, Tianxi Cai, Junwei Lu.

http://jmlr.org/papers/v24/22-0642.html

#embeddings #embedding #factorization

We are happy to announce that a new thematic programme with 5 focus weeks just got started at
@ESIVienna
! Check out the details below

31st July - 1st September 2023
Schrödinger Lecture Hall

Programme description
http://www.esi.ac.at/events/e476/

Subject: #QuantumFieldTheory at the Frontiers of the #StrongInteraction
Week 1: Finite-Mass and #Electroweak Effects in #gaugetheories
Week 2: #Singularity Structure of Quantum Field Theory Beyond the Leading Power
Week 3: #Factorization Violation and the #Space of Universal Functions
Week 4: Simulation of the All Order Structure of Scattering Amplitudes
Week 5: Multi-Variable Techniques for All Order Resummations in QFT

(see the motion picture at https://twitter.com/ESIVienna/status/1686002264878530561)

@univienna

One day, one decomposition
A025283: Composites that use the same digits as their prime factorization

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A025283.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A025283.html

One day, one decomposition
A022559: Sum of exponents in prime-power factorization of n!

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A022559.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A022559.html