O problema parece ser que provavelmente o crontab não conhece o PATH onde o comando arp mora.
Eu usaria:
* * * * * /usr/sbin/arp -n >> results.txt
No entanto, eu usaria o arpwatch para monitorar as alterações do ARP. Funciona como um daemon e registra as alterações do MAC em um arquivo ao longo do tempo, juntamente com o tempo de época da alteração. Também é capaz de enviar mensagens para syslog e e-mails.
Arpwatch keeps track for ethernet/ip address pairings. It syslogs activity and reports certain changes via email. Arpwatch uses pcap(3) to listen for arp packets on a local ethernet interface.
Report Messages
Here's a quick list of the report messages generated by arpwatch(1) (and arpsnmp(1)):
new activity This ethernet/ip address pair has been used for the first time six months or more.
new station The ethernet address has not been seen before.
flip flop The ethernet address has changed from the most recently seen address to the second most recently seen address. (If either the old or new ethernet address is a DECnet address and it is less than 24 hours, the email version of the report is suppressed.)
changed ethernet address The host switched to a new ethernet address.
Syslog Messages
Here are some of the syslog messages; note that messages that are reported are also sysloged.
ethernet broadcast The mac ethernet address of the host is a broadcast address.
ip broadcast The ip address of the host is a broadcast address.
bogon The source ip address is not local to the local subnet.
ethernet broadcast The source mac or arp ethernet address was all ones or all zeros.
ethernet mismatch The source mac ethernet address didn't match the address inside the arp packet.
reused old ethernet address The ethernet address has changed from the most recently seen address to the third (or greater) least recently seen address. (This is similar to a flip flop.)
suppressed DECnet flip flop A "flip flop" report was suppressed because one of the two addresses was a DECnet address.
Files
/var/lib/arpwatch - default directory
arp.dat - ethernet/ip address database
ethercodes.dat - vendor ethernet block list