A400000 Some possible candidates

[Ed Pegg]

I went looking for recent sequences I could explain well. 

A395424 Busy Beaver problem for FRACTRAN   
A395745 Initial digit of 3^(3^(3^n))  
A396001 Worst greedy Egyptian-fraction length for denominator n  
A396924 Connected regular integral graphs on n vertices  
A399136 Smallest polycube with an n-cell cavity  
A399138 Maximum no-3-in-line set in an n x n x n grid       

I'd like to see other candidate lists... unless the mods say no.  

[Sean A. Irvine]

If people want to discuss their candidates or offer up lists like Ed's for consideration, then that is fine by me.

Sean.

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[Geoffrey Caveney]

I like

A395013 Deficient numbers k for which the k-th Fibonacci number is abundant.

225, 315, 525, 675, 735, 855, 1125, 1155

Alexander Violette has found additional terms 1485, 1995, 9975. Presumably other terms less than 1485 have not yet been ruled out.

Maximilian Hasler asks whether all terms of this sequence will be multiples of 15.

The related sequence A074726, of all numbers k for which the k-th Fibonacci number is abundant, dates back to 2002, and still the largest term in the accompanying b-file is only 1422 (updated in 2022).

Geoffrey

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[Dave Consiglio]

Of course I like this one: https://oeis.org/draft/A398712 - Numbers n such that all of the digits of n are not represented in the Collatz pathway from n to 1. 

But my honorable mention is  https://oeis.org/A398007 - a(n) is the largest k such that the vertices of a k X k grid can be painted in n colors without having any 3 collinear vertices of the same color.

I like sequences that are easy for non-experts to understand, are fairly short and computationally complex. Ripe for new terms and potentially related sequences.

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[Charles Greathouse]

At a glance I do like

A399138 a(n) = maximum number of points in the n X n X n grid with no three of them collinear.

(similar to A398007 that Dave mentioned) as well as

A396829 a(n) is the least number of prime factors for any squarefree abundant number with prime(n) (the n-th prime) as its least prime factor.

but I haven't had a chance to look through in a principled way yet. I don't intend to nominate any sequence I (co)authored, but

A395235 Powerful numbers of the form k^2 - 2.

is surprisingly nice with an unobvious clean upper bound.

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[Bence Hervay]

Well, A395424 is by far the "least arbitrary" out of these, and therefore mathematically beautiful in my opinion; most other nominations I've seen either:

- have many similar versions, and I wouldn't want to feel like "why not that other version" (like why not allow this kind of edge, why exactly 3 dimensions, why exactly this recurrence, or why not choose a different starting case, etc)

- are specific to a decimal base

A395424 is beautiful because:

- FRACTRAN is one of the mathematically simplest systems that are capable of universal computation

- despite the definition of program length being a bit arbitrary, determining the terms for any sensible metric is equally difficult!

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[M. F. Hasler]

Bence Hervay wrote:

Well, A395424 is by far the "least arbitrary" out of these, (...) 

most other nominations I've seen either:

- have many similar versions, and I wouldn't want to feel like "why not that other version" 

- are specific to a decimal base

A395424 is beautiful because:

- FRACTRAN is one of the mathematically simplest systems that are capable of universal computation

Yes, I agree that FRACTRAN is a great concept.

I have added e.g. Python & PARI code for PRIMEGAME in OEIS.org/A203907 and also 

A395539 : Successor function of Conway's PIGAME.

[ oh - I could nominate that ! :-D ]

[ Also, I and/or others should elaborate on PIGAME:
AFACS this (defn of the successor function) and the two sequences with numerators & denominators of the FRACTRAN program, are the only place where PIGAME is mentioned in OEIS! ]

But as you say:

- despite the definition of program length being a bit arbitrary, (...)

and in view of that, isn't A395424 just an "arbitrary variant" of

A060843 = 1, 6, 21, 107, 47176870, ... : Busy Beaver problem 

?

Which looks like a subsequence of the FRACTRAN variant up to 107, 

but then it has 47176870 which is not in A395424,

while A395424 has the larger 31957632 as first more-then-3-digits term.

Why is that?  It is intriguing to me!

- Maximilian

[Ed Pegg]

Agree with many points ... especially similar to others, specific base. 
This has both faults.   

A395745 Initial digit of 3^(3^(3^n))  

2, 7, 1, 1, 1, 1, 1, 8, 6, 1, 1, 2, 1, 1, 5, 6, 1, 1, 2, 1  

But it does look bizarre so far.  I like weird behavior.

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[Geoffrey Caveney]

I observe that A300000 (which is interesting!) is specific to a decimal base, so I would prefer that A400000 not also have that property.

But if one could produce the first *n* digits of 3^(3^(3^n) for the terms of A395745, it might be worthy :)

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