Functional Puzzle Solving

A professor has written a note to help him remember his password. I'm sure you can find out what it is. After all, he's very functional.

$\text{4,24 AOPCRTLIFA}$
$\text{10,5 RAVHLE}$
$\text{0,1 SIOEICN}$
$\text{3,27 BEUMC}$
$\text{54,18 ETEITNTO}$
$\text{100,2 LHOIGAMCTR}$
$\text{16,4 UOATRSROOEQ}$
$\text{0,-1/2 UATZE}$
$\text{6,18 IELPRNT}$
$\text{140,12 MSGTAI}$

$\text{My password:}$
$\text{41,2,72,21,12,42,2,71,25,6,2,39}$

MATHEMATICAL

Here's how to find it.

Each set of letters anagram a mathematical function that can be applied to the first number to produce the second. (I actually didn't realize the relationship between the number until I had found most of them by recognizing near-anagrams).  4,24 FACTORIAL + P 10,5 HALVE + R 1,0 COSINE + I 3,27 CUBE + M 54,18 TOTIENT + E 100,2 LOGARITHM + C 16,4 SQUARE ROOT + O 0,-1/2 ZETA + U 6,18 TRIPLE + N 140,12 SIGMA + T

From here,

The extra letters spell "PRIME COUNT". Applying the prime count function (number of primes $\leq n$) to each number gives $12, 0, 19, 7, 4, 12, 0, 19, 8, 2, 0, 11,$ which as alphabet letters spells MATHEMATICAL.

• Great puzzle! The whole process felt smooth and justified. I like the recursive step to get the final answer. I have some really nitpicky nitpicks. Since the ordering of the letters to anagram is arbritary (unless I'm missing something), convention is to write them alphabetically. I'd find a 1-indexed alphabet a bit more natural than 0-indexed. Commas are a bit weird to represent functions, though maybe $a \to b$ would be a giveaway, but if so the final clue should avoid commas and be space-separated. – xnor Apr 15 '15 at 8:30
• To my knowlege, the last 'sub-puzzle' is one indexed, not zero indexed (?). Take 2. The number of primes <= 2 is one, so 2 -> 1 -> 'A'. But you're right, in that the solution phrase should probably not be separated by commas, and you also picked that I tried not to use -> to avoid giving the game away too early. – Tryth Apr 15 '15 at 8:37
• @Tryth My mistake, it was indeed one-indexed. – xnor Apr 15 '15 at 8:39
• @xnor : 100,2 LOGARMITH + C should be LOGARITHM – Tim Couwelier Apr 15 '15 at 13:24
• LOGARITHM -> ALGORITHM – Joe Z. Apr 16 '15 at 0:22