[amoringello]

The use of WI-FI is fully explained on page 88 of the K-1 manual, including situations when WI-FI is disabled.

Could you expand on what you're referring to? Why do people need to be so difficult?
Two situations are listed; when USB is connected and when in Movie mode... and while powered off, of course.

Less text than explaining where to find it. I can see helping someone to research their own problems, but the reference is not at all helpful because it is quite obvious that the information does not even apply.

P88:
----
This camera has a built-in Wi-Fi function. By directly connecting to a communication device such as smartphones and tablets via wireless LAN, the camera can be operated from the communication device and images can be shared with others via social networks or e-mail.

Enabling the Wi-Fi Function
The Wi-Fi function is disabled when the camera is turned on. Enable the function using one of the following methods.
• Settingfromthemenus(p.89)
• SettingwiththeSmartFunction(p.89)

When the Wi-Fi function is enabled, the Wi-Fi lamp lights up and the 7 (white) icon indicating the wireless LAN status is displayed on the status screen and Live View screen. When 8 (gray) is displayed, a connection with the communication device is not established correctly.

t Memo
• Even after the Wi-Fi function is enabled, it returns to an inactive
state when the camera is turned off and on again. Also, when
the camera enters C mode, the Wi-Fi function is disabled.
• Auto Power Off is not available while a Wi-Fi connection is established or images are being transferred. If Auto Power Off is activated while a connection is not established or during standby, the Wi-Fi function is disabled. The function is enabled when the camera recovers from Auto Power Off.
• TheWi-Fif unction is disabled during the USB connection.

u Caution
• DonotattempttousetheWi-Fifunctioninalocationwhere
the use of wireless LAN devices is restricted or prohibited,
such as on airplanes.
• WhenusingtheWi-Fifunction,observelocallawsand
regulations governing radio communication.
• Ifamemorycardwithabuilt-inwirelessLANfunction(suchasan Eye-Fi card or a Flucard) is used, radio wave interference may occur. When this occurs, disable the Wi-Fi function of the camera.

[rawr]

could I use a HooToo Wireless Router connected by a USB cable to a portable hard drive to copy K-1 photos via K-1's wireless system WITHOUT using a computer, iPad or computer?

I do remember that some WiFi SD cards, like the Toshiba FlashAir or EyeFi Mobi, promise to do something automated with your WiFi image transfers, as this review of that EyeFi card suggests:

'Lastly, you can use Eyefi's desktop software to give the card access to a home or studio wireless network. When transferring directly to a phone, tablet or computer, the card automatically takes over your device's Wi-Fi connection, sends the images over and then drops the connection.

So it may be worth exploring what some of those WiFi SD cards can do, if you really need a kind of autonomous file transfer capability across several devices on your network.

While I am on the subject of Pentax WiFi, it is also interesting to note that one of the biggest WiFi SD card brands - the Toshiba FlashAir - is designed to work with Ricoh's Keenai automated photo storing/sharing service.

Keenai is the only service that downloads your images directly from cameras* and makes them available automatically on all your devices.

https://www.keenai.com

And the latest Toshiba FlashAir card supports it directly

the latest FlashAir™ works with Ricoh Innovations Corporation’s ‘Keenai’ photo storage and management service. Keenai allows the automatic transfer of photo data from FlashAir™ cards and enables users to autosync photo and movie data to the cloud. Details of Keenai can be found at: Keenai | Photo app for cameras, iPhone, android and desktop.

Funny that Pentax camera WiFi seems unaware of this service, whilst Toshiba's WiFi card directly integrates with it.