Riffs and Rotes

[Jon Awbrey]

Cf: Riffs and Rotes • Happy New Year 2023 !!!
https://inquiryintoinquiry.com/2023/01/01/riffs-and-rotes-happy-new-year-2023/

All,

Let pₙ = the n-th prime.

Then 2023
= 7∙17²
= p₄∙p₇²
= p_{p₁^p₁} ∙ {p_{p₄}}^{p₁}
= p_{p₁^p₁} ∙ {p_{p_{p₁^p₁}}}^{p₁}

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

2023 = p_{p^p} ∙ {p_{p_{p^p}}}^p

Forms like those correspond to a family of digraphs
called “riffs” and a family of graphs called “rotes”,
as described in the following OEIS Wiki article.

Riffs and Rotes ( https://oeis.org/wiki/Riffs_and_Rotes )

The riff and rote for 2023 are shown in the next two Figures.

Riff 2023
https://inquiryintoinquiry.files.wordpress.com/2023/01/riff-2023-big-1.jpg

Rote 2023
https://inquiryintoinquiry.files.wordpress.com/2023/01/rote-2023-big-1.jpg

Regards,

Jon ( https://mathstodon.xyz/@Inquiry )

[Jon Awbrey]

Riffs and Rotes • Happy New Year 2024
• https://inquiryintoinquiry.com/2024/01/02/riffs-and-rotes-happy-new-year-2024/

Let pₙ = the n-th prime.

Then 2024
= 2³∙11∙23
= p₁³∙p₅∙p₉
= p₁^p₂ ∙ p_p₃ ∙ p_p₂²
= p₁^p_p₁ ∙ p_p_p₂ ∙ p_p_p₁^p₁
= p₁^p_p₁ ∙ p_p_p_p₁ ∙ p_p_p₁^p₁

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

• 2024 = = p^p_p ∙ p_p_p_p ∙ p_p_p^p

The article referenced below tells how forms like that

correspond to a family of digraphs called “riffs” and

a family of graphs called “rotes”.

The riff and rote for 2024 are shown in the next two Figures.

Riff 2024
• https://inquiryintoinquiry.files.wordpress.com/2024/01/riff-2024.png

Rote 2024
• https://inquiryintoinquiry.files.wordpress.com/2024/01/rote-2024.png

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

Happy New Year!

Jon

cc: https://www.academia.edu/community/lK1anV