1, 1.8, 3.3, 6.2, 12, 22, 43, 82, 160, 300, 560, 1K1, 2K0, 3K9, 7K5, 13K, 24K, 47K, 91K, 180K, 330K, 620K, 1.2M, 2.2M, 4.3M, 8.2M.
Ok, so how do you plan to set 1R4 to 1% with those values?
I don't. Let me see if I can clarify:
With the arrangement above (with a switchable relay 'shorting' each resistor when contacts are closed), you can set the box to any ohm value +- appx 1ohm. All the resistors are in series, so with all of the relays open, you get the sum of all of the resistor values. With all of the resistors closed, you get whatever the contact resistance of all of the relays are. So, you effectively you can set the box to a nominal value from something around 1ohm, to something over 10Mohm.
This is not the accuracy of the box - it's the resolution, or in ADC terms, step size. You can't get 1R4, but you can definitely set to a value you believe is something like 23.837K+-1Ohm. To do this, you need the actual measured resistance of the box when all relays are closed, and then the resistance delta for each relay when opened individually - which will be close to the value of the resistor in parallel with the contact. Once you have those values, writing a bit of code to adjust the relays appropriately to make any value by opening the correct relays is rather trivial.
Which gets us back to the accuracy question.
Below about 100 ohms you're limited by the appx 1 ohm step size.
Above 100 ohms you're limited by drift effects of the resistors+relay+copper+everything else. Trying to set a box made from 50ppm resistors to something like 10,000,000 ohms and expecting it to be +- 1 ohm isn't likely, since 1 degree of temperature drift will net you 500 ohms of change. But expecting 1% at the top end of the scale is more than reasonable. Probably more like 0.1% (but don't quote me on that as it would need to be characterized/simulated).
In relation to the goal, it's mostly these two:
a) cover whole range, with any value from the range, to 1%,
c) cover unspecified count of randomly distributed values but not using more than 26 relays and 34 m of wire,
With the caveat of the low end of the scale is going to be swamped by the step size, and the number of relays+resistors is still to be determined.
or an additional option: Buy something off the shelf which simulates a real resistor.