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Consider the function $f(n, i)$ which returns the exponent of the $p_i$ in the factorisation of $n$, where $p_i$ is the $i$-th prime.

Question: is there a standard label for $f$?

Context: In the first edition of my Gödel book, I (thoughtlessly!) used the notation $\mathit{exp}(n, i)$, with 'exp' for 'exponent'. But of course that notation invites the misreading '$n$ to the power $i$' (taking 'exp' for exponential). Ooops! I'd like to do better in the second edition. Suggestions?

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    It does not address the numbering of the primes, but i like $v_p(n),$ being the exponent of $p$ in factoring $n.$ This is called the $p$-adic valuation, see Gouvea's book. I suppose most use $\mbox{ord}_p(n).$ You could make something from either that includes your $i.$2012-07-20

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In a grad course many years ago, from a student of Kleene, the notation $(n)_i$ was used. I believe at that time the notation was quite standard, at least in the English-speaking parts of North America. For whatever it's worth, the notation is used here, in the long list of basic primitive recursive functions.

Seems fine for a logic course, since the notation is relatively short-term, while one is proving the basic facts about the indexing.

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    Some people use this to mean the falling factorial (http://en.wikipedia.org/wiki/Pochhammer_symbol).2012-07-20
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    @QiaochuYuan: Not the only instance of multiple context-dependent meanings!2012-07-20
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    On the other hand, it doesn't seem likely that the Pochhammer symbol would turn up in the context of elementary number theory, so as long as there's the customary "Let blah be the foo of..." sentence somewhere in the beginning, it ought to be fine.2012-07-21
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    Thanks. I should indeed have mentioned the $(n)_i$ notation, which we inherit from Kleene's _Introduction to Metamathematics_, p.230. I must say that I've never found it either pretty in use or memorable, which is no doubt why I avoided it in my book. But I guess the Wikipedia entry indicates that it _is_ standard notation if anything is.2012-07-21
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Perhaps "ord" might be a potential alternative, as it seems to fit with the definition used in $p$-adic numbers. In some books I am reading, $ord_p(n)$ would be the greatest power of $p$ which divides $n$, so maybe $ord_p(n,i)$ might be suitable. However, you might already be using this notation for something else?

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    Some people use $\text{ord}_p(n)$ to mean the order of $n$ in the unit group of $\mathbb{Z}/p\mathbb{Z}$.2012-07-20