#OEIS #Partitions
https://oeis.org/A334442
https://oeis.org/A334439

How should integer partitions be listed in the OEIS, what offset should the sequences have, and how should the empty partition be handled?

As an example, let's consider A334439: Irregular triangle whose rows are all integer partitions of n sorted first by sum, then by length, ...

This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.

0
(1)
(2)(11)
(3)(21)(111)
(4)(31)(22)(211)(1111)
(5)(41)(32)(311)(221)(2111)(11111)

The apex of this triangle is the empty partition, which, however, does not appear in the sequence as it is represented in the OEIS.

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, ..

Probably therefore the index was set to 1. However, this is disputed; the original author had set it to 0. Which value do you consider correct? The same applies to A334301 and A334302.

And then there's also A334442: Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, ...

This sequence can also be interpreted as the following triangle:

0
(1)
(2)(11)
(3)(12)(111)
(4)(13)(22)(112)(1111)
(5)(14)(23)(113)(122)(1112)(11111)

Listed as: 1, 2, 1, 1, 3, 1, 2, 1, 1, .

Here, the offset has been set to 0! Do you see any reason to set it differently in this case than in the other three? Or is it clear to you that A334442 has offset 0, and A334439 has offset 1?

[Q 1] Don't you expect the same offset in all cases?
[Q 2] What offset do you expect? 0 or 1?

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Someone asked me about sexy cousins. I'll admit, I was briefly hopeful. Turns out they meant primes with gap 6 and gap 4 respectively.

Disappointed, I did what anyone would do and stayed up computing distinct-prime partitions of primes.

Define m_d<(p) as the minimum k such that prime p is a sum of k pairwise distinct primes, each strictly less than p. Set m_d<(p) = ∞ if no such decomposition exists.

For every prime p ≤ 10^8 (verified directly, ~30 seconds on commodity hardware):

• m_d<(11) = ∞, uniquely. 11 simply refuses to be built from smaller primes. Provable by exhaustion: the 11 subsets of {2,3,5,7} of size ≥ 2
sum to {5,7,8,9,10,12,14,15,17}. 11 is not invited.

• m_d<(17) = 4, uniquely. 17 needed all four of its juniors (2+3+5+7) just to show up. Overachiever or socially awkward — you decide.

• m_d<(p) ∈ {2, 3} for every other prime in range. 5,761,448 tested. All well-adjusted.

I don't know if this object has a name. I don't know if this is trivially known, or trivially reducible to something known.

The repo has code, a writeup, verification prompts you can feed to any LLM to check the claims, and some other things I found along the way.

https://github.com/keeltremor/goldeen
https://doi.org/10.5281/zenodo.19542143

If any of this rings a bell please let me know!
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Linux Partition Types

Primary: Bootable partitions, max 4 per disk
Extended: Special container that holds logical partitions
Logical: Reside inside extended, unlimited count

MBR: Old standard, 2TB limit, 4 primary max
GPT: Modern, >2TB disks, 128+ partitions, UEFI-ready

Choose GPT for new systems. Your future self thanks you. 💾

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