OFFSET
0,10
COMMENTS
This sequence, together with A345228, provides information regarding generalized Collatz functions. (Replace 3*k+1 in the standard Collatz function with a more general a*k+b; then a = 1+2*n and b = 1+2*A345228(m,n).) A345228 tells us which m are part of a cyclic orbit but not if these are part of the same cycle. This sequence identifies each distinct cycle with a different number. Example: If A345228(m1,n) = A345228(m2,n) we know m1 and m2 are part of a cycle but not necessarily the same cycle. If T(m1,n) <> T(m2,n) we know m1 and m2 are not in the same cycle.
The value of n appears to have only a small effect in this sequence and in a majority of cases we find T(m,n) = A000265(m) holds true. This is surprising, given how n is involved in the definition.
LINKS
FORMULA
T((1+2*n)*m,n)/T(m,n) = 1+2*n.
T((1+2*(n-b))*m,n)/T(m,n) = 1+2*(n-b). 0 <= b <= n. This formula is only for the majority of cases true if b > 0. For each column m are some rows n where an exception will be seen.
T(m,n) <= A000265(m) (largest odd divisor of m).
T(m,n) = A000265(m) For the majority of all n.
EXAMPLE
Twelve initial terms of rows 0-10 are listed below:
n |m->
0: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 1, 11, ...
1: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, ...
2: 0, 1, 1, 1, 1, 5, 3, 7, 1, 3, 5, 1, ...
3: 0, 1, 1, 3, 1 5, 3, 7, 1, 9, 5, 11, ...
4: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
5: 0, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 11, ...
6: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 1, ...
7: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
8: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
9: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
10: 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, ...
Example: T(3,4) = 3 -> f(n): k/2; (9*k+21)/2. This is because r = A345228(3,4) = 10 and 2*10+1 = 21.
f(3) = 24, f(24) = 12, f(12) = 6, f(6) = 3, f(3) = 24, ....
The smallest number in this cycle is 3.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Scheuerle, Jun 14 2021
STATUS
approved