# [Jelly], 18 bytes

    J:g¥Ƭ’}ṪU×ḷb§¥ƬṪS%

[Try it online!][TIO-lzokdrap]

[Jelly]: https://github.com/DennisMitchell/jelly
[TIO-lzokdrap]: https://tio.run/##y0rNyan8/9/LKv3Q0mNrHjXMrH24c1Xo4ekPd2xPOrQcJAbkB6v@P7xcXzPy///o6GhDHQUTHQXDWB1Do1gdruhocx1DHSMdYx0THVMdMx1zHQsdSx1DAx1LuAIgByqPKWQBFDKECEGMJcJkQwPsRkPMgYgYw0ViAQ "Jelly – Try It Online"

A less naive computation of the digital roots comes out a bit longer by having to special-case 0. Inverted output: truthy (nonzero) if the checksum fails.

```
J                     1-indices from the start
 :                    divided by
  g¥                  their GCDs with
     ’}               B - 1
    Ƭ  Ṫ              repeated while unique,
        U             reversed
         ×ḷ           times each N.
              ƬṪ      Loop while unique:
           b ¥        Convert each to base B
            §         and sum each.
                S     Sum
                 %    mod B.
```

A version which ties with an alternative to repeated division:

# [Jelly], 18 bytes

    :gɓ’*
    JçU×ḷb§¥ƬṪS%

[Try it online!][TIO-lzokw588]

[Jelly]: https://github.com/DennisMitchell/jelly
[TIO-lzokw588]: https://tio.run/##y0rNyan8/98q/eTkRw0ztbi8Di8PPTz94Y7tSYeWH1p6bM3DnauCVf8fXq6vGfn/f3R0tKGOgomOgmGsjqFRrA5XdLS5jqGOkY6xjomOqY6ZjrmOhY6ljqGBjiVcAZADlccUsgAKGUKEIMYSYbKhAXajIeZARIzhIrEA "Jelly – Try It Online"

`ç` is used to save 1 byte compared to an excessive combination of quicks in `:g¥*@¥’}` or `:g¥’*¥@¥`:

```
:        Divide the indices by
 g       their GCDs with
  ɓ’     B - 1
    *    raised to the power of each index.
```

No prime factor of any index has a multiplicity greater than that index, so it suffices to exponentiate B - 1 that many times.