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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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    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