A389673 - OEIS

A389673

a(n) is the length of the sequence obtained by repeated iteration of the "rebase-and-read" operation (A389674), starting at n until one reaches a number already in the sequence.

1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 9, 9, 5, 6, 8, 10, 5, 11, 1, 10, 4, 9, 7, 5, 3, 10, 10, 1, 11, 9, 4, 9, 11, 2, 7, 5, 1, 8, 12, 4, 10, 9, 6, 10, 10, 1, 7, 5, 5, 9, 12, 5, 10, 9, 1, 13, 6, 2, 7, 8, 5, 7, 5, 1, 9, 14, 6, 6, 6, 4, 7, 9, 1, 7, 6, 6, 7, 11, 2, 8, 2, 1, 7

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OFFSET

1,2

COMMENTS

See A389674 for the "rebase-and-read" operation.

Fixed points occur when A007953(n) = 10, that is elements of A052224.

It is not known if a(n) is finite for all n. Empirically, a(n) appears to be always finite and grows without bound, albeit slowly, as n grows.

The only cycle known with length greater than one has length four, arising from 115 --> 223 --> 436 --> 277.

No further cycles for n <= 10^10. - Michael S. Branicky, Oct 12 2025

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000

EXAMPLE

For n = 13, we successively apply A389674 obtaining the sequence 13 --> 31 --> 133 --> 250 --> 505 --> 505. Since we hit a repeated number on the 5th iteration, a(13) = 5.

a(15) = 8 and is the earliest point at which we encounter a repeating cycle: 15 --> 23 --> 43 --> 61 --> 115 --> 223 --> 436 --> 277 --> 115.

MATHEMATICA

A389674[n_] := If[# == 1, n, FromDigits[IntegerDigits[n, #]]] & [Total[IntegerDigits[n]]];

A389673[n_] := Length[NestWhileList[A389674, n, UnsameQ, All]] - 1;

Array[A389673, 100] (* Paolo Xausa, Oct 17 2025 *)

PROG

(Python)

from sympy.ntheory import digits

def F(n): return n if (b:=sum(d:=list(map(int, str(n)))))<2 else sum(di*10**i for i, di in enumerate(digits(n, b)[:0:-1]))

def a(n):

c, seen = 0, set()

while n not in seen: n, c, seen = F(n), c+1, seen | {n}