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Given a sequence $A=(a_1,\ a_2,\ \ldots)$ one can define the records of $A$ as numbers $a_n:n\in\mathbb{Z}^+$ such that $a_n>a_m$ whenever $n>m.$ So you start at 1 and write down every number larger than all preceding numbers. (Of course you could also look for record-small numbers in just the same way.)

Is there a standard term for the terms $a_n:n\in\mathbb{Z}^+$ such that $a_n

These are somewhat more tricky to work with since you can't prove membership by checking finitely many values. But they are often defined and useful. Surely there is a standard term for this somewhere; I'm loathe to invent terminology except when absolutely necessary.

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    How about high-water and low-water marks?2012-07-31
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    @Théophile: To me these suggest the two kinds of records in the first paragraph, where you're walking from 1 upward and looking for the smallest yet or the largest yet.2012-07-31
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    What you call *records* are sometimes called *champions* in the Number Theory literature. But I can't say I've ever come across a term for the other concept.2012-08-01
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    @Charles Ah, right. I spent a few minutes just now thinking about a good prefix for high/low-water mark, but arrived at nothing but barbarisms.2012-08-01
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    @Théophile: I appreciate that, thanks for your time.2012-08-01

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There is precedent for the term anti-champion. Bukor, Filakovszky, and Toth, On the diophantine equation $x_1x_2\cdots x_n=h(n)(x_1+x_2+\cdots+x_n)$, Ann Math Silesiannae 12 (1998) 123-130, available here, write on page 128,

Denote by $f(n)$ the number of solution [sic] of $$x_1x_2\cdots x_n=n(x_1+x_2+\cdots+x_n),\qquad x_1\le x_2\le\cdots\le x_n.$$ A number $n$ is called a champion if $f(n)\gt f(m)$ for every $m\lt n$, it is called [an] anti-champion if for every $m\gt n\quad f(m)\gt f(n)$. It is true that the anti-champions are always primes?