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I am looking for a term for numbers that have a base of $2$ with any power so for example, $2,4,8,16,32,\cdots$.

I would say a base $2$ number but am under the assumption that that refers to binary numbers. My best idea so far is a power of $2$ but I'm looking for something more elegant or more simple than power of $2$.

Any ideas?

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saying a number is a "power of 2" is pretty elegant all by itself – zerosofthezeta Aug 15 '13 at 18:48
If you want to call a bus top a xilt, that is fine, particularly in a private diary. Just don't expect someone to respond when you ask where the nearest xilt is. – André Nicolas Aug 15 '13 at 19:08
OP: For your interest, the accepted answer is currently the single most downvoted accepted answer of the whole MSE site. – Did Aug 24 '13 at 8:20

Those are called "powers of $2$", or possibly "perfect powers of $2$". I believe there is no other common name for them.

Here's the OEIS entry for them. OEIS calls them "powers of $2$".

Don't call them "base-$2$ numbers"; nobody will know what you mean, and everyone will think you mean something else.

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How about "even powers of $2$" :-) – Stefan Hamcke Aug 15 '13 at 18:30
Like I said, I am looking for something more elegant than "power of 2". Even if we have to make one up as long it makes sense. – jhello Aug 15 '13 at 18:39
@Stefan that could also mean "powers of 4". Looking at the google results for "odd power of 2" and "even power of 2" it seems like when people say that what they really mean is "power of 2 with even/odd exponent". The word "power" seems to be used rather vaguely. – Dan Brumleve Aug 15 '13 at 18:40
@joelliusp I think your opinion of the "elegance" of the name is outweighed here by the value of calling them by the same name that everyone else uses. Relevant. – MJD Aug 15 '13 at 18:46
@DanBrumleve: This would resemble the "odd primes" denoting primes larger 2, and here the "even powers of 2" refer to powers of 2 larger 1 :) – Stefan Hamcke Aug 15 '13 at 18:49

First power of 2: $$2^1=2$$(read "two to the first power")

Second power of 2: $$2^2=2*2$$(read "two to the second power")

Third power of 2: $$2^3=2*2*2$$(read "two to the third power")

Fourth power of 2: $$2^4=2*2*2*2$$ (read "two to the fourth power")

nth power of 2: $$2^n= \underbrace{2*2*2*\ldots*2}_{\text{$n$ factors}}$$

(read "two to the nth power")

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The sequence $2,4,8,16,32,64,...$ is known referred to as the binary sequence. So perhaps "element of the binary sequence" will fit your requirements suffice given adequate context.

EDIT: Clearly, the set $B$ of all persons using the term binary sequence for $1,2,4,8,...$ is non-empty. The book linked in MJD's comment is one I taught from for many years, and I suspect many students and teachers have encountered this book. I will admit that this author did stretch several conventions in order to appeal to his unsophisticated audience, but I am certain this is not the only place I have seen this usage.

This appears to be yet another case in mathematics where a term is used differently in different contexts (the word "normal" being perhaps the most abused term in all of mathematics). Anon's comment and my Google searching reveal that the set $B$ I mention seems to be a distinct minority amongst a super-set of all persons who use the term 'binary sequence' in any way.

However, I argue that this does not justify saying that this usage is non-standard or idiosyncratic. Merely less common.

So getting back to the OP's question, you can use my original answer, but you will have to ensure your audience understands your meaning.

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@cobaltduck: I don't think that's true. – Ryan Budney Aug 15 '13 at 19:14
Is your best evidence that the phrase "binary sequence" is "known" (your word) to refer to powers of two really an obscure SO question with an idiosyncratic usage? It is rather easy to research and find (if one isn't already aware) that the phrase actually is known to refer to a sequence of binary digits, not the sequence of powers of two, so this answer is obviously wrong. – anon Aug 16 '13 at 8:18
@cobaltduck Very curious that this was the OP's accepted answer. – Did Aug 16 '13 at 11:23
@anon As I pointed out above, Google searches (web) (books) easily reveal that cobaltduck's answer is correct and that the term is widely used, not just in "one obscure SO question". – MJD Aug 16 '13 at 13:46
@MJD Somehow I admire your enterprise of rehabilitation (probably for its quixotic side), but I am afraid I cannot share its conclusions. The absolute number of occurrences of uses of the term with the quite idiosyncratic meaning you advocate is irrelevant if the number of occurrences with the usual meaning dwarfs it--and it most certainly does, hence your repeated assertions that "the term is widely used" or that "the answer is correct" are just wrong. If there is nothing better than X, the answer to "give me something better than X" must NOT be that "Y is better than X". Sorry. – Did Aug 16 '13 at 17:19

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