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PowerShell, 97 bytes param($n)$a,$b,$c=$n|group;(($n-join'*'|iex),(($a,$b|?{$_.count-eq1}).name,1)[$a.count-eq3])[!$c] Try it online! Not terribly impressive, but given that PowerShell doesn't h …

PowerShell, 67 bytes param([uint64]$n)($n/[math]::Sqrt(2)).ToString("G17")-replace'\..*' Try it online! … However, the [math]::Sqrt() function only works on Double, so there we're stuck. So far, so standard. …

PowerShell, 64 59 bytes for($a="$args";9-lt$a){$a;$a="$(($a|% t*y)-join'*'|iex)"}$a Try it online! Iterative method. Takes input and stores it into $a, then enters a for loop so long as the leng …

PowerShell, 107 bytes param([bigint]$a)for([bigint]$b=1;-join("$([bigint]::pow($a,$b))"|% t*y|sort -u)-ne-join(0..9);$b=$b+1){}$b Try it online! Pretty straightforward, just a shame we need to u …

PowerShell, 68 56 bytes param($n)1..$n|%{$a+=$_*$_*!($n%$_)};1..$a|?{$_*$_-eq$a} Try it online! Seems long ... -12 bytes thanks to mazzy Does exactly what it says on the tin. Takes the range fr …

$n|%{$o+=[math]::pow($x,$_)};$o Try it online! Not bad for needing a .NET call for pow. -7 bytes thanks to mazzy. …

PowerShell, 72 bytes param($x,$y)$x..$y|%{$o+=([convert]::ToString($_,2)-replace0).length};$o Try it online! Long because of the conversion to binary [convert]::ToString($_,2) and getting rid of …

PowerShell, 121 bytes param($a)$b="";($a|%{if($_-ge25){$b+="$_ "}else{$b;$b=""}})+$b|?{(-split$_).count-ge5-and(-split$_|?{$_-ge30}).count-ge3} Try it online! or Verify all test cases PowerShell …

Specifically, Conway's PRIMEGAME. This is an algorithm devised by John H. Conway to generate primes using a sequence of 14 rational numbers: A B C D E F G H I J K L M N 17 …

Given an input integer n >= 10, output the average of all deduplicated rotations of the integer. For example, for input 123, the rotations are 123 (no rotation), 231 (one rotation) and 312 (two rotat …

PowerShell, 96 84 bytes $y=-(([char[]]($x="$args"-replace'^978|.$')|%{--$a*[int]"$_"})-join'+'|iex)%11;$x+$y Try it online! Takes input "$args", does a regex -replace to get only the pertinent p …

PowerShell, 87 96 bytes param($a,$b)filter f($n){(1..($n-1)|?{!($n%$_)})-join'+'|iex}(f $a)-eq$b-and(f $b)-eq$a-and$a-$b Try it online! Takes input $a,$b. Defines a filter (here equivalent to a …

PowerShell, 57 bytes param($n)(2*$n-1)..++$n|?{'1'*$_-match'^(?!(..+)\1+$)..'} Try it online! Takes input $n, constructs a range from 2*$n-1 to ++$n (i.e., excluding the endpoints), and feeds th …

PowerShell, 40 bytes param($a)0..9|?{!(($a-replace'x',$_)%9)} Try it online! or Verify all test cases Takes input like '123x' into $a. Constructs a range 0 to 9 and uses Where-Object (here abbre …

PowerShell, 110 bytes $a=,0*101;2..100|%{$i=$_;for($z=$j=0;++$j-lt$i;$z+=$k-eq1){for($k=$j;$j%$k-or$i%$k;$k--){}};($a[$i]=$a[$z]+1)} Try it online! Mathematical approach. Actually, looking thro …