Growth and trajectories in an additive digit map

[Arnav T]

Axxxxxx Growth and trajectories in an additive digit map: a(n) is the n-th iterate of T starting from seed 321, where T(m) = L(m) + S(m), with L(m) the digits of m in descending order and S(m) in ascending order (leading zeros allowed).

(Temporary placeholder number until assignment.)

NAME

Growth and trajectories in an additive digit map: 

iterates of the additive Kaprekar-type process T(n) = L(n) + S(n).

DEFINITION

For a decimal integer n with 3 digits (allowing leading zeros), let L(n) be the digits of n sorted in descending order and S(n) the digits sorted in ascending order.

  T(n) = L(n) + S(n).

Let a(0) = 321, and for k ≥ 1, a(k) = T(a(k−1)).

DATA

321, 444, 888, 1776, 9438, 13332, 45654, 110100, 111111, 222222, 444444, 888888, 1777776, 9554439, 13333221, 45566442, 110199999, 1222333321, 3444555666, 7888888888, 17777777776, 95555554319, …

OFFSET
0,1

COMMENTS

• This is the additive analogue of Kaprekar’s routine (which subtracts ascending from descending digits).
• Only the seed 000 leads to the trivial fixed point 0.
• Other seeds (e.g., 321 or 102) quickly grow into large numbers, often passing through repdigits (like 111, 222, 444, 888) that act as quasi-attractors.
• Unlike Kaprekar’s process, there is no small nontrivial fixed point for 3-digit seeds; almost all trajectories diverge.
• Exhaustive computation for seeds 000–999 shows only seed 000 stabilizes at 0.

REFERENCES

• D. R. Kaprekar, Mathematical Recreations, on Kaprekar routines.
• J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations.

EXAMPLE
Starting with seed a(0) = 321:
 a(1) = 321 + 123 = 444,
 a(2) = 444 + 444 = 888,
 a(3) = 888 + 888 = 1776,
 a(4) = 976 + 679 = 9438, etc.

Starting with seed a(0) = 102:
 a(1) = 210 + 012 = 222,
 a(2) = 222 + 222 = 444, then continues as above.

FORMULA

T(n) = L(n) + S(n), where L(n) is the digits of n in descending order and S(n) in ascending order (leading zeros allowed).

CROSSREFS
Cf. A099009 (Kaprekar process for 3-digit numbers, constant 495),
Cf. A099010 (Kaprekar’s 4-digit constant 6174),
Cf. A033865 (reverse-and-add sequences).

KEYWORD

nonn,base,easy,new,look,unexplored

AUTHOR

Arnav Taras , Sep 2025

STATUS

new, awaiting review

Can some one add this to OEIS, if correct .. Idk how to update the entry.

[Giovanni Resta]

A few observations in random order.

The instruction for submitting a new sequence are clearly stated here: https://oeis.org/Submit.html

Follow them.

Before submitting a new sequence or submitting an edit to an old sequence it is MANDATORY to 

read very carefully the StyleSheet for OEIS https://oeis.org/wiki/Style_Sheet

and also to check that the sequence or trivial modifications of the sequence are not already in OEIS.

Your sequence is wrong starting from term following 1777776, that should be 9455538, not 9554439.

This makes me suspect you computed the sequence by hand, which in general is a very bad idea.

The completely arbitrary starting point (321) would probably disqualify this sequence from being added to OEIS.

A reasonable starting point for this kind of sequence is simply 1.

The process is not limited to 3-digit numbers, why insisting on that detail?

Names of sequences should  be precise and possibly short. In your case 

the preamble "Growth and trajectories in an additive digit map:" adds nothing.

The fact that there are no fixed points apart 0 is obvious and there is no need of "extensive computations": 

by construction T(n) > L(n) >= n for every n>0, because  if you put the digits of a number in decreasing order 

you cannot obtain a number smaller than the original number.

Giovanni

[Arnav T]

Mam Giovanni,

Thank you very much for carefully reviewing my draft and pointing out the mistake after 1777776, as well as the issues with the arbitrary seed, naming, and formatting. You are absolutely right — I computed some terms by hand, which was a mistake, and I will switch to programmatic verification.

I will also rework the submission to follow the OEIS Style Sheet  more closely, remove the arbitrary starting point, and keep the definition short and precise.

I appreciate your guidance and patience.

Best regards,
Arnav Taras