A357954 - OEIS

%I #63 Dec 02 2022 07:05:28

%S 1,2,3,4,13,18,28,118,194,289,338,353,354,419,489,528,609,769,1269,

%T 1299,2081,4890,4891,9113,18575,18702,20759,35084,1874374,338749352,

%U 2415951874

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

%C 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.

%C The sequence is finite.

%C 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.

%C 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).

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_digital_invariant">Perfect digital invariant</a>

%e k=9113 is a fixed point (perfect digital invariant) for the reiterated function A357143(k):

%e     9113_10 = 242423_5 (a 6-digit number in base 5);

%e     A357143(9113) = 2^6 + 4^6 + 2^6 + 4^6 + 2^6 + 3^6 = 9113.

%e k=18702 is a sociable digital invariant for the reiterated function A357143(k), requiring 2 iterations:

%e   1st iteration:

%e     18702_10 = 1044302_5 (a 7-digit number in base 5);

%e     A357143(18702) = 1^7 + 0^7 + 4^7 + 4^7 + 3^7 + 0^7 + 2^7 = 35084;

%e   2nd iteration:

%e     35084_10 = 2110314_5 (also a 7-digit number in base 5);

%e     A357143(35084) = 2^7 + 1^7 + 1^7 + 0^7 + 3^7 + 1^7 + 4^7 = 18702.

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

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

%K nonn,base,fini,full

%O 1,2

%A _Francesco A. Catalanotti_, Oct 22 2022