A357954 Integers k that are periodic points for some iterations of k->A357143(k).

1, 2, 3, 4, 13, 18, 28, 118, 194, 289, 338, 353, 354, 419, 489, 528, 609, 769, 1269, 1299, 2081, 4890, 4891, 9113, 18575, 18702, 20759, 35084, 1874374, 338749352, 2415951874

Offset: 1

Author

Francesco A. Catalanotti, Oct 22 2022

Keywords

• nonn
• base
• fini
• full

Comments

Given the function A357143(k), a number k is a term of the sequence if there exists a j such that A357143^j(k) = k, where j is the number of iterations applied.

The sequence is finite.

Proof: A357143(k) < k for all big enough k. g(k) = A110592(k)*4^A110592(k) is clearly an upper bound of A357143(k). Hence k > g(k) -> k > A357143(k), therefore every periodic point must be in an interval [s;t] such that for every k in [s;t] k <= g(k). Limit_{k->oo} g(k)/k = 0; now using the little-o definition we can show that there always exists a certain k_0 such that, for every k > k_0, k > g(k). The conclusion is that there must exist a finite number of intervals [s;t] and, consequently, a finite number of periodic points.

Every term k of the sequence is a periodic point (either a perfect digital invariant or a sociable digital invariant) for the function A357143(k).

The longest cycle needs 6 iterations to end: [489, 609, 769, 1269, 1299, 2081].

Examples

			k=9113 is a fixed point (perfect digital invariant) for the reiterated function A357143(k):
    9113_10 = 242423_5 (a 6-digit number in base 5);
    A357143(9113) = 2^6 + 4^6 + 2^6 + 4^6 + 2^6 + 3^6 = 9113.
k=18702 is a sociable digital invariant for the reiterated function A357143(k), requiring 2 iterations:
  1st iteration:
    18702_10 = 1044302_5 (a 7-digit number in base 5);
    A357143(18702) = 1^7 + 0^7 + 4^7 + 4^7 + 3^7 + 0^7 + 2^7 = 35084;
  2nd iteration:
    35084_10 = 2110314_5 (also a 7-digit number in base 5);
    A357143(35084) = 2^7 + 1^7 + 1^7 + 0^7 + 3^7 + 1^7 + 4^7 = 18702.
		

Links

• Wikipedia, Perfect digital invariant

Crossrefs

Cf. A357143 , A010346 (fixed points), A110592 (exponents p(k)).

Cf. A157714 (base-10 sociable digital invariants), A101337 (A357143(k) base 10), A151544 (base-10 periodic points).