Francesco A. Catalanotti - cp's OEIS frontend

User: Francesco A. Catalanotti

Francesco A. Catalanotti has authored 2 sequences.

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

Francesco A. Catalanotti, Oct 22 2022

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].

			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.

Wikipedia, Perfect digital invariant

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).

A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 5, 10, 17, 4, 5, 8, 13, 20, 9, 10, 13, 18, 25, 16, 17, 20, 25, 32, 1, 2, 9, 28, 65, 2, 3, 10, 29, 66, 9, 10, 17, 36, 73, 28, 29, 36, 55, 92, 65, 66, 73, 92, 129, 8, 9, 16, 35, 72, 9, 10, 17, 36, 73, 16, 17, 24, 43, 80, 35, 36, 43, 62, 99, 72, 73, 80, 99, 136, 27

Offset: 1

Francesco A. Catalanotti, Oct 26 2022

			For n = 13_10 = 23_5 (2 digits in base 5): a(13) = 2^2 + 3^2 = 13.
For n = 73_10 = 243_5 (3 digits in base 5): a(73) = 2^3 + 4^3 + 3^3 = 99.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A357954, A010346, A110592.

Cf. in base 10: A157714, A101337, A151544.

f:= proc(n) local L,d,i;
  L:= convert(n,base,5);
  d:= nops(L);
  add(L[i]^d,i=1..d)
end proc:
map(f,[$1..100]); # Robert Israel, Oct 26 2023

a[n_] := Total[IntegerDigits[n, 5]^IntegerLength[n, 5]]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

a(n) = my(d=digits(n, 5)); sum(k=1, #d, d[k]^#d); \\ Michel Marcus, Oct 29 2022

from sympy.ntheory.factor_ import digits
def A357143(n):
    t = len(s:=digits(n,5)[1:])
    return sum(d**t for d in s) # Chai Wah Wu, Oct 31 2022

a(n) = Sum_{i=1..A110592(n)} d(i)^A110592(n), where d(i) is the i-th digit of n in base 5.

Corrected and extended by Michel Marcus, Oct 29 2022

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User: Francesco A. Catalanotti

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

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A357143 a(n) is sum of the base-5 digits of n each raised to the number of digits of n in base 5.

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