Riffs and Rotes • Happy New Year 2026

[Jon Awbrey]

Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/

Let p_n = the n-th prime.

Then 2026 = 2 ∙ 1013

= p_1 p_170

= p_1 p_{2 ∙ 5 ∙ 17}

= p_1 p_{p_1 p_3 p_7}

= p_1 p_{p_1 p_{p_1} p_{p_4}}

= p_1 p_{p_1 p_{p_1} p_{p_{{p_1}^{p_1}}}}

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

2026 = p p_{p p_p p_{p_{p^p}}}

The article linked below tells how forms of that order
correspond to a family of digraphs called “riffs” and
a family of graphs called “rotes”.

• https://oeis.org/wiki/Riffs_and_Rotes

The riff and rote for 2026 are shown in the next two Figures.

Greetings, SeqFans, here is the Rote for 2026 ...

• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.png

Anyway, I think I got that right ...

Rushed for time today so I'll do the Riff tomorrow ...

Regards,

Jon

[Jon Awbrey]

typo, of course, I think the graph is correct ...

Jon

On 1/1/2026 5:16 PM, Jon Awbrey should have written:

> Riffs and Rotes • Happy New Year 2026
> • https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/
>
> Let p_n = the n-th prime.
>
> Then 2026 = 2 ∙ 1013
>
>           = p_1 p_170
>
>           = p_1 p_{2 ∙ 5 ∙ 17}
>
>           = p_1 p_{p_1 p_3 p_7}
>

>           = p_1 p_{p_1 p_{p_2} p_{p_4}}
>
>           = p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}

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

> 2026 = p p_{p p_{p_p} p_{p_{p^p}}}

[Jon Awbrey]

Riffs and Rotes • Happy New Year 2026
• https://inquiryintoinquiry.com/2026/01/01/riffs-and-rotes-happy-new-year-2026/

Seqfans,

Here's the corrected and completed version
(apologies for delay, came down with a cold)

Let p_n = the n-th prime.

Then 2026 = 2 ∙ 1013

= p_1 p_170

= p_1 p_{2 ∙ 5 ∙ 17}

= p_1 p_{p_1 p_3 p_7}

= p_1 p_{p_1 p_{p_2} p_{p_4}}

= p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}}

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

2026 = p p_{p p_{p_p} p_{p_{p^p}}}

The article linked below tells how forms of that order
correspond to a family of digraphs called “riffs” and
a family of graphs called “rotes”.

• https://oeis.org/wiki/Riffs_and_Rotes

The riff and rote for 2026 are shown in the next two Figures.

Riff 2026
• https://oeis.org/wiki/File:Riff_2026_Card.png
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/riff-2026-card.png

Rote 2026
• https://oeis.org/wiki/File:Rote_2026_Card.png
• https://inquiryintoinquiry.com/wp-content/uploads/2026/01/rote-2026-card.png

Regards,

Jon