New OEIS Sequences: Monster Moonshine Exponents (15 sequences from A001379

[Jim Dupont]

Dear OEIS Editors,

I would like to submit 15 new integer sequences derived from the Monster group irreducible representations, related to Monstrous Moonshine.

The list is attached and here in gist, one seq per SSP.
https://gist.githubusercontent.com/jmikedupont2/ee9748143fdbdc70266fc9fa690ea52d/raw/7d67a9ccb92c99b036feb2e6554f7e0680bcefc4/oeis_monster_ssp_.txt

BACKGROUND:
The Monster group is the largest sporadic simple group (order |M| ≈ 8.08×10^53). It has 194 irreducible representations with dimensions given in OEIS A001379. The j-invariant modular form coefficients decompose as weighted sums of these dimensions - this is the McKay-Thompson observation that inspired Monstrous Moonshine (Conway & Norton 1979, proven by Borcherds 1992).

THE SEQUENCES:
For each of the 15 supersingular primes p (dividing |M| with (p+1) | 24), we define:
  a(n) = v_p(A001379(n)) = exponent of p in the n-th Monster irreducible representation (n = 0 to 193)

Where v_p(x) is the p-adic valuation. Each sequence has exactly 194 terms.

- exp_2 through exp_71 (one for each SSP prime: 2,3,5,7,11,13,17,19,23,29,31,41,47,59,71)

FORMULA:
For each sequence: a(n) = v_p(A001379(n))

LINK TO A001379:
These sequences are derived from and directly Cross-Reference OEIS A001379 (dimensions of Monster irreps). They provide the p-adic valuations for each SSP prime.

INTERESTING FINDINGS:

1. **196883 = 47 × 59 × 71**: The smallest non-trivial Monster irrep (A001379(1) = 196883) has exactly these three largest SSP primes as factors with exponent 1 each. This was McKay's original observation (noting that 196884 = 196883 + 1).

2. **j-coefficient decompositions** (McKay-Thompson):
   - j₁ = 196884 = A001379(0) + A001379(1) = 1 + 196883
   - j₂ = 21493760 = A001379(0) + A001379(1) + A001379(2) = 1 + 196883 + 21296876
   - This shows j-invariant coefficients are weighted sums of Monster irreps

3. **Defect Zero Representations**: For larger SSP primes (41, 47, 59, 71), the preponderance of "1" entries mark representations where v_p(dimension) = 1 - these are the "defect zero" representations that remain irreducible when reduced modulo p.

4. **Quotient Relationship**: Every prime factor in all 194 irreps divides |M|. Since dim(irrep) | |M| for any finite group, these exponent sequences are bounded by the exponents in |M|.

5. **Spiral / 12-turns**: The 12-turn spiral visualization corresponds to 1/2 × Leech lattice dimension (24).

REFERENCES:
- OEIS A001379: Degrees of irreducible representations of Monster group M
- Monstrous Moonshine: Conway & Norton (1979), Borcherds (1992)

DATA LINKS (Public GitHub Gists):
- OEIS sequence data (JSON): https://gist.github.com/jmikedupont2/9e179632200019619c01a1a07cc91b80
- Matrix heatmap visualization: https://gist.github.com/jmikedupont2/c350d5684246918be7dd39a5b9663694
- LLM brief generator: https://gist.github.com/jmikedupont2/b0766375056f3e0c86ae7357902e809a

PROPOSED NAMES (example for prime 2):
- A00???(n) = exponent of 2 in dimension of n-th Monster irreducible representation

Please advise on A-numbers and any corrections.

Best regards,
J. Mike Dupont

[Sean A. Irvine]

Hi Jim,

Submissions to the OEIS need to go through the web interface.

Assuming you don't already have an account (and I could not see one), the first step is to register for an OEIS account, that can be done here:

https://oeis.org/wiki/Special:RequestAccount

Once you have an account you can pre-allocate the A-numbers for your sequences, but note you will be limited to making 3 submissions at a time.

For more information on the overall process, please see:

https://oeis.org/wiki/Overview_of_the_contribution_process

Regards,

Sean.

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[Jim Dupont]

sure thing, thanks!

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[Jim Dupont]

https://oeis.org/draft/A394826 I will just do one first

[Allan Wechsler]

Jim, if I'm understanding these correctly, they are all finite 193-element sequences, and you know all the elements. If that's the case, remember to mark the sequences "fini" and "full".

-- Allan

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[Fred Lunnon]

<< (dividing |M| with (p+1) | 24) >>

I don't think this is true for p = 71 , say! WFL

_

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[Fred Lunnon]

<< (dividing |M| with (p+1) | 24) >>

On closer inspection, this quotation seems weirdly unrelated to context.

The submission in general seems curiously full of unnecessary detail,

while lacking references which might serve to explain its terminology.

Frankly, my suspicions are aroused ...

WFL

_

[Jim Dupont]

I can rework this guys I try to couple different ones it shouldn't be that hard to explain it's just a prime factorization and splitting of the other sequence

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[Jim Dupont]

I have updated the entry to make it simpler. thank you for your feedback this is my first time contributing.