A359221 - OEIS

%I #18 Jan 05 2023 18:55:58

%S 0,1,2,3,12,28,227,821,22246,26494,204953,425720

%N Starting numbers which reach a new record high value when iterating the map x->A359194(x) (binary complement of 3n).

%C It is unknown whether all starting numbers reach 0.

%C 103721 is not a term of this sequence despite having a trajectory of record length, because its maximum of 2.42...*10^14081 is lower than the previous record holder.

%H Joshua Searle, <a href="https://qedscience.wordpress.com/2021/02/19/collatz-inspired-sequences/">Collatz-inspired sequences</a>

%e Let S(x) = iteration sequence of A359194 starting with x; then

%e S(0) = (0), maximum = 0;

%e S(1) = (1, 0), maximum = 1;

%e S(2) = (2, 1, 0), maximum = 2;

%e S(3) = (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0), maximum = 300;

%e Since S(3) contains a higher maximum than any lower positive starting integer, 3 is a term of this sequence.

%o (Python)

%o from itertools import count, islice

%o def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)

%o def itersmax(n):

%o     i, fi, m = 0, n, n

%o     while fi != 0: i, fi, m = i+1, f(fi), max(m, fi)

%o     return i, m

%o def agen(): # generator of terms

%o     record = -1

%o     for m in count(0):

%o         v, mx = itersmax(m)

%o         if mx > record:

%o             yield m # use mx to obtain values

%o             record = mx

%o print(list(islice(agen(), 8))) # _Michael S. Branicky_, Dec 22 2022

%Y Cf. A035327, A359194, A359207, A359208, A359209, A359215, A359218, A359219, A359220, A359222, A359255.

%K nonn,base,more,hard

%O 1,3

%A _Joshua Searle_, Dec 22 2022