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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?

jhello
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2 Answers2

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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.

Stefan Hamcke
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MJD
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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
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    @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
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    @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
  • @Stefan I agree that it makes more sense that way. Also I would read "odd prime power" the same way (a power of a prime other than 2), and "even prime power" would never be uttered (just say "power of 2" or "square of a prime power" depending). But "odd power of 2" always means "twice a power of 4". Strange, right? – Dan Brumleve Aug 15 '13 at 18:54
  • @DanBrumleve: I don't remember how I came up with "even power of 2" in the first place. I think someone suggested "positive power of 2" in an answer now deleted, implying that "power of 2" could allow each integer (positive or negative) as exponent. But I think "power of 2" is the most common and it is understood that the sequence starts with $2,4,...$, so "even" is not really needed. – Stefan Hamcke Aug 15 '13 at 19:03
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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")

leo
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