The first 15 terms of the sequence `{a_i} = 2^i`

are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. All of the digits in base-10, i.e. {0, 1, 2, ..., 9}, each appear in at least one of these terms, and 32768 is the first term to contain the digit 7.

Define the *digital potency* (this is a term I came up with, I am unaware if there is a proper term for this) of *n* in base-10 to be the minimum value of *x* such that the digits {0, 1, 2, ..., 9} each appear in at least one of *n*, *n^2*, *n^3*, ..., *n^x*.

Thus, from the above example, the digital potency of 2 in base-10 is 15.

More generally, define the digital potency of *n* in base-*b* to be the minimum value of *x* such that the digits {0, 1, 2, ..., b-1} each appear in at least one of the base-*b* representations of *n*, *n^2*, *n^3*, ..., *n^x*.

**Observation 1:** For any numeral system base-*b*, we can construct a number with digital potency of 1 by simply concatenating the digits {0, 1, 2, ..., b-1}.

**Observation 2:** If *n* is a perfect power of *b*, then the digital potency of *n* in base-*b* is undefined or 'infinite' (i.e., there exists a digit in {0, 1, 2, ..., b-1} that never appears in any of the terms of the sequence `{a_i} = m^i`

). I believe that this is actually true so long as *n* is a rational power of *b*.

Has this concept been studied before? I wrote some Python code to calculate the digital potency of various numbers in various bases, and it seems to have some interesting behavior. For instance, if you fix *n* and calculate the digital potency for various choices of *b*, the values seem rather unpredictable. Similarly, if you fix *b* and calculate the digital potency for various choices of *n*, the values also seem unpredictable.

Some specific questions I have:

- Is there an easy way to calculate or at least approximate digital potency without explicitly calculating the powers of a number?
- Is there a non-trivial example of a number and a base system where digital potency is undefined?

neveroccurs as first digit. $\endgroup$ – Gerry Myerson Apr 10 at 3:23seconddigit. The Weyl equidistribution law still applies. $\endgroup$ – Bullet51 Apr 10 at 4:26