Hidden Pandigitals

When we think of pandigital numbers, it is a number like 1263480759 that comes to mind. Each of the digits from 0 to 9 occur exactly once. There are 3,265,920 such numbers (leading zeroes not being considered) out of the total of 3,486,784,401 possible ten digit numbers. That's a representation of about 0.094%.

There are however, other ways in which pandigital numbers can arise and one way is in the decimal approximations of the square roots of whole numbers when approximated to ten digits NOT ten decimal places. Some of the digits will occur in the whole number part of the square root and the rest will occur in the decimal part. An example is the square root of 1362 where we have:$$ \sqrt{1362}=36.90528417 \dots$$Note that the final digit arises from truncation of the infinite decimal and not from rounding. Essentially, the decimal point is ignored and so associated with certain whole numbers are their pandigital square roots expressed as whole numbers themselves with the decimal points ignored. The complete list of such numbers, up to 40000, is as follows:$$ \begin{align} 1362 &\rightarrow 3690528417\\1843 &\rightarrow 4293017586\\2540 &\rightarrow 5039841267\\4280 &\rightarrow 6542170893\\5507 &\rightarrow 7420916385\\6896 &\rightarrow 8304215796\\14601 &\rightarrow 1208345976\\15143 &\rightarrow 1230568974\\17547 &\rightarrow 1324650897\\18393 &\rightarrow 1356207948\\20337 &\rightarrow 1426078539\\22710 &\rightarrow 1506983742\\23560 &\rightarrow 1534926708\\25887 &\rightarrow 1608943752\\27487 &\rightarrow 1657920384\\30728 &\rightarrow 1752940386\\32286 &\rightarrow 1796830542\\32615 &\rightarrow 1805962347\\33144 &\rightarrow 1820549367\\34499 &\rightarrow 1857390642\\37194 &\rightarrow 1928574603 \end{align} $$Of course it's easy enough to see where the decimal point should be. These numbers form OEIS A113507 (permalink). We can extend this idea to cube roots and in so doing the first number to make an appearance is 2017 because:$$ (2017)^{1/3}=12.63480759 \dots$$The numbers up to 40000 are (permalink):$$ \begin{align} 2017 &\rightarrow 1263480759\\3053 &\rightarrow 1450693287\\9950 &\rightarrow 2150837964\\15139 &\rightarrow 2473806519\\15533 &\rightarrow 2495083671\\18357 &\rightarrow 2637954108\\24214 &\rightarrow 2893047156\\24424 &\rightarrow 2901386574\\31457 &\rightarrow 3156742089\\32654 &\rightarrow 3196284750\\39605 &\rightarrow 3408657291 \end{align} $$These numbers form OEIS A119517. This approach can be extended to fourth roots and beyond. We can also consider whole numbers raised to let's say \(1/ \pi \). For example:$$3638^{1/ \pi}=  13.59746028 \dots$$jUpt to 40000, the following whole numbers lead to pandigitals when raised to the power \(1/ \pi\):$$ \begin{align} 3638 &\rightarrow 1359746028\\7109 &\rightarrow 1682940375\\10271 &\rightarrow 1892064735\\11572 &\rightarrow 1965273840\\13818 &\rightarrow 2079431586\\14435 &\rightarrow 2108547963\\20539 &\rightarrow 2359047168\\20981 &\rightarrow 2375089614\\26220 &\rightarrow 2549731608\\27158 &\rightarrow 2578419036\\27313 &\rightarrow 2583094176\\35022 &\rightarrow 2795816403\\35330 &\rightarrow 2803619574\\35901 &\rightarrow 2817964035\\37023 &\rightarrow 2845703691 \end{align} $$Not surprisingly this sequence of numbers is not found in the OEIS.