numbers (version 0.8-5)

collatz: 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)

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.

Arguments

n

integer to start the Collatz sequence with.

k, l

parameters for computing k*n+l.

short

logical, abbreviate stps with (k*n+l)/2

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.

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

if (FALSE) {
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
}