Re: [SeqFan] Subject: New Sequence Proposal: The Ng. Randiv Digit-Product Offset Sequence (with Cycle-7 Attractor)

[brad klee]

Hi Ng. Randiv, 

Don't name a sequence after yourself. Do give the reader motivation why 

the sequence matters. 

You seem to want a Collatz sequence, but I don't think this one is it, because

it should be easy to prove that trajectories don't have interesting heights:

Let XY...Z be some number, the product of digits is maximized for all digits

equal to 9. The base case 9*9 = 81 < 9*10 = 90 < 99 leads into an induction 

where at the next step 9*9*9 = 729 < 9*10*10 = 900 < 999, etc, etc.  

It's then easy to see that XY...Z can not go and get more digits, it can only lose 

digits. When it does, either it either loops on level or loses more digits. 

This isn't the behavior, for example, of A00884: https://oeis.org/A008884 , which 

has a trajectory with interesting heights and complexity gain. 

We already have A007954. This doesn't add much more, so I don't know 

why your submission should be included. You can try to defend it, or maybe 

just the digits squared sequence if that's not already recorded. I would suggest 

go back to drawing board and look for something else. 

If you want to go in the Collatz direction--which I wouldn't necessarily recommend 

unless you think maybe new tools will help--you need to look for an iterator that 

increases and decreases whatever height function you like, maybe digit count. 

I recommend choosing problems that have a mixture of numbers and geometry. 

Geometry is also important in all areas of science. For example, look here: 

https://en.wikipedia.org/wiki/Logistic_map

Somewhere on this page there must be an opportunity to find something new and

interesting to do. As an added benefit to this study, once you know the difference 

between logistic and exponential it should be easier for you to discern bogus 

singularity news that doesn't take costs into account. 

All the best, 

--Brad

 

 

On Monday, July 13th, 2026 at 7:13 AM, NG. Randiv Singha <kranjit...@gmail.com> wrote:

[Geoffrey Caveney]

I think the function is more interesting in base 4, where 10, 11, 12, 13 (that is, 4, 5, 6, 7 in base 10) each maps to itself. This raises the interesting question of which other integers, upon iterated mapping of this function in base 4, terminate in each of the values 10, 11, 12, 13.

It appears that no integer other than 11 itself maps to 11.

The following integers terminate in 10 upon iterated mapping: 1, 2, 3, 10, 20, 22, 23, 30, 32, 111, 112, 113, 121, 131, 211, 223, 232, 311, 322, ...

The following integers terminate in 12 upon iterated mapping: 12, 21, 100, 101, 102, 103, 110, 120, 130, 133, 200, 201, 202, 203, 210, 220, 222, 230, 300, 301, 302, 303, 310, 313, 320, 330, 331, 333, 1000, ...

Integers terminating in 13 upon iterated mapping appear to occur with significantly less frequency, but they do exist: 13, 31, 33, 122, 123, 132, 212, 213, 221, 231, 233, 312, 321, 323, 332, 11111, 11112, ..., 12222, ..., 22222, ..., 111112, 111113, ..., 111133, ..., 111233, ..., 1111222, ...

It appears that almost all powers of the base 4 terminate in either 10 or 12 upon iterated mapping, and thus also almost all integers containing a digit 0 in base 4 representation.

However, there is at least one example of a power of the base 4 that terminates in 13 upon iterated mapping:

10^210 (4^36 = 4,722,366,482,869,645,213,696 in base 10) has 211 digits, and 211^2 = 111121. This number has digit product 2 and 12 digits, and 2+12^2 = 212. This has digit product 10 and 3 digits, and 10+3^2 = 31. This has digit product 3 and 2 digits, and 3+2^2 = 13.

It appears that such examples must be very rare, and it seems to me to be an interesting open question whether there are finitely many or infinitely many such powers of the base 4 that terminate in 13 upon iterated mapping of this function. A brief search up to 10^2000 (4^128) does not find even a second example yet. The large gap in the set of smaller integers that terminate in 13, between 332 (62 in base 10) and 11111 (341 in base 10), may suggest a similarly large gap between 10^210 (4^36) and the next smallest example, if indeed another example exists.

Geoffrey

On Mon, Jul 13, 2026 at 8:13 AM NG. Randiv Singha <kranjit...@gmail.com> wrote:

Dear SeqFan Community,

I have recently submitted a new sequence draft to the OEIS and would highly appreciate your insights, algorithmic verification, or programming checks.

The sequence is officially titled:
The Ng. Randiv Digit-Product Offset Sequence

It is defined by the following discrete arithmetic operator:
R(n) = (product of digits of n) + (number of digits of n)^2

Data (the first 40 terms for n = 1 to 40):
2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 4

Behavioral Characteristics:
1. Dynamic Trajectory: While the initial terms mapped above represent the first step for consecutive integers, iterating the function on any single starting integer seed causes the trajectory to rapidly collapse into a stable, 7-stage periodic attractor loop:
6 -> 7 -> 8 -> 9 -> 10 -> 4 -> 5 -> 6.

2. The 'Zero-Slam' Boundary Condition: For exceptionally large numbers containing the digit 0, the digit-product vanishes completely. The value immediately drops down to the baseline offset value k^2 (where k is the number of digits), causing an aggressive, fast-acting collapse toward the baseline loop.

I am a 16-year-old student from Assam, India, and this is my first formal mathematical submission. I would love to see if anyone in the community can generate further terms, write code scripts (PARI/GP, Python, Mathematica) for the database, or analyze maximum trajectory heights before collapsing into the loop.

Best regards,
Ng. Randiv Singha
St. Capitanio Senior Secondary School, Silchar, Assam, India

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[brad klee]

Base 2 isn't going to do much. I don't know if Base 3 is any better than Base 4. 

One way to find out is just to make the call back tree. 

Here is experimental data from Harm.On.ica (now GPT Sol 5.6):

<<

call back trees for: 

    https://oeis.org/A007954

BASE 3

TREE 0

0

├── 10

├── 20

├── 100

├── 101

├── 102

├── 110

├── 120

├── 200

├── 201

├── 202

├── 210

├── 220

├── 1000

├── 1001

├── 1002

├── 1010

├── 1011

├── 1012

└── 1020

TREE 1

1

├── 11

│   ├── 22

│   │   ├── 222

│   │   ├── 1222

│   │   ├── 2122

│   │   ├── 2212

│   │   └── 2221

│   ├── 122

│   ├── 212

│   ├── 221

│   ├── 1122

│   ├── 1212

│   ├── 1221

│   ├── 2112

│   ├── 2121

│   └── 2211

├── 111

├── 1111

└── 11111

TREE 2

2

├── 12

├── 21

├── 112

├── 121

│   ├── 2222

│   ├── 12222

│   ├── 21222

│   ├── 22122

│   └── 22212

├── 211

├── 1112

├── 1121

├── 1211

├── 2111

├── 11112

├── 11121

├── 11211

├── 12111

└── 21111

BASE 4

TREE 0

0

├── 10

│   ├── 22

│   ├── 122

│   └── 212

├── 20

├── 30

├── 100

├── 101

├── 102

├── 103

├── 110

├── 120

├── 130

├── 200

├── 201

├── 202

├── 203

├── 210

└── 220

TREE 1

1

├── 11

├── 111

├── 1111

├── 11111

├── 111111

├── 1111111

├── 11111111

├── 111111111

├── 1111111111

├── 11111111111

├── 111111111111

├── 1111111111111

├── 11111111111111

├── 111111111111111

├── 1111111111111111

├── 11111111111111111

├── 111111111111111111

├── 1111111111111111111

└── 11111111111111111111

TREE 2

2

├── 12

│   ├── 23

│   ├── 32

│   ├── 123

│   │   └── 333

│   ├── 132

│   ├── 213

│   ├── 231

│   ├── 312

│   └── 321

├── 21

│   ├── 33

│   ├── 133

│   ├── 313

│   └── 331

├── 112

├── 121

├── 211

└── 1112

TREE 3

3

├── 13

├── 31

├── 113

├── 131

├── 311

├── 1113

├── 1131

├── 1311

├── 3111

├── 11113

├── 11131

├── 11311

├── 13111

├── 31111

├── 111113

├── 111131

├── 111311

├── 113111

└── 131111

>>

On Monday, July 13th, 2026 at 4:45 PM, Geoffrey Caveney <geoffre...@gmail.com> wrote:

I think the function is more interesting in base 4 . . .