Lmst

One day, one decomposition
A000028: Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #primes #factorization #PrimeNumbers #binary #expansions #graph #threejs #webGL

Decomposition into weight × level + jump of A000028 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

Decomposition into weight × level + jump of A000028 in 2D (log(weight), log(level))

One day, one decomposition
A276078: Numbers n in whose prime factorization no exponent of any prime(k) exceeds k

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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #factorization #exponent #primes #PrimeNumbers #graph #threejs #webGL

Decomposition into weight × level + jump of A276078 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A276078 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A260047: Composites whose prime factorization in base 3 is an anagram of the number in base 3

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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #prime #factorization #anagram #graph #threejs #webGL

Decomposition into weight × level + jump of A260047 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A260047 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

One day, one decomposition
A260046: Composites whose prime factorization in base 2 is an anagram of the number in base 2

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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #composites #prime #factorization #anagram #graph #threejs #webGL

Decomposition into weight × level + jump of A260046 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A260046 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

A first successful factorization of RSA-2048 integer by D-Wave quantum computer

https://www.sciopen.com/article/10.26599/TST.2024.9010028

#HackerNews #quantumcomputing #RSA2048 #DWave #factorization #cybersecurity #technews

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

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Decomposition into weight × level + jump of A242414 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A242414 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

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
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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #PrimeNumbers #graph #threejs #webGL

Decomposition into weight × level + jump of A212168 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212168 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

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
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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #graph #threejs #webGL

Decomposition into weight × level + jump of A212166 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212166 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

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
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#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #exponent #prime #factorization #graph #threejs #webGL

Decomposition into weight × level + jump of A212165 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212165 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

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
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Decomposition into weight × level + jump of A212164 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A212164 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

'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

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Decomposition into weight × level + jump of A130091 in 2D (log(weight), log(level))

Decomposition into weight × level + jump of A130091 in 3D (threejs - WebGL) (log(weight), log(level), log(jump))

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