| collatz {numbers} | R Documentation |
Collatz Sequences
Description
Generates Collatz sequences with n -> k*n+l for n odd.
Usage
collatz(n, k = 3, l = 1, short = FALSE, check = TRUE)
Arguments
n |
integer to start the Collatz sequence with. |
k, l |
parameters for computing |
short |
logical, abbreviate stps with |
check |
logical, check for nontrivial cycles. |
Details
Function n, k, l generates iterative sequences starting with
n and calculating the next number as n/2 if n is
even and k*n+l if n is odd. It stops automatically when
1 is reached.
The default parameters k=3, l=1 generate the classical Collatz
sequence. The Collatz conjecture says that every such sequences will end
in the trivial cycle ...,4,2,1. For other parameters this does not
necessarily happen.
k and l are not allowed to be both even or both odd – to make
k*n+l even for n odd. Option short=TRUE calculates
(k*n+l)/2 when n is odd (as k*n+l is even in this case),
shortening the sequence a bit.
With option check=TRUE will check for nontrivial cycles, stopping
with the first integer that repeats in the sequence. The check is disabled
for the default parameters in the light of the Collatz conjecture.
Value
Returns the integer sequence generated from the iterative rule.
Sends out a message if a nontrivial cycle was found (i.e. the sequence is not ending with 1 and end in an infinite cycle). Throws an error if an integer overflow is detected.
Note
The Collatz or 3n+1-conjecture has been experimentally verified
for all start numbers n up to 10^20 at least.
References
See the Wikipedia entry on the 'Collatz Conjecture'.
Examples
collatz(7) # n -> 3n+1
## [1] 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
collatz(9, short = TRUE)
## [1] 9 14 7 11 17 26 13 20 10 5 8 4 2 1
collatz(7, l = -1) # n -> 3n-1
## Found a non-trivial cycle for n = 7 !
## [1] 7 20 10 5 14 7
## Not run:
collatz(5, k = 7, l = 1) # n -> 7n+1
## [1] 5 36 18 9 64 32 16 8 4 2 1
collatz(5, k = 7, l = -1) # n -> 7n-1
## Info: 5 --> 1.26995e+16 too big after 280 steps.
## Error in collatz(5, k = 7, l = -1) :
## Integer overflow, i.e. greater than 2^53-1
## End(Not run)