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January 11

Numbers that are products of single powers of primes

Google is failing me. I am looking for the name of and/or information about numbers whose prime decomposition only has single powers. The series goes 6, 10, 14, 15, 21, 30, etc, and does not include 12=3*2^2, 9=3^2, etc. I forget why I want it, but it's in my notes. Anybody familiar with it? SamuelRiv (talk) 18:00, 11 January 2014 (UTC)[reply]

I believe OEIS: A120944 is what you're looking for. --Kinu ^t/_c 18:40, 11 January 2014 (UTC)[reply]

That's it! The square-free integers -- I think one of the reasons I was looking at them was because a student of mine asked if there was a rule about what numbers are allowed under a fully-simplified square root and what are not. The other reason was probably something to do with power sets. SamuelRiv (talk) 04:55, 12 January 2014 (UTC)[reply]

The list in your original question isn't quite the square-free integers, as you've omitted the primes (and 1). The sequence Kinu linked to, which matches your list, is the square-free composite (positive) integers. AndrewWTaylor (talk) 23:56, 12 January 2014 (UTC)[reply]

Set of all functions whose degree is 0

Look at the Degree of a polynomial article.

It has a section that talks about the degrees of functions that are not polynomials.

The first 2 are easy to see the meaning of. They extend the degree to all real numbers as opposed to just the non-negative integers.

However, let's play with the statement that the degree of $logx$ is 0.

This statement reveals that sometimes, a function that is not a constant can have a degree of 0.

Given this statement, we can create functions with logarithms for any degree by multiplying a polynomial by the logarithmic function, and using this rule it does not change the degree. For example, the degree of $x^{5}*logx$ is 5.

Is there any rule we can use to create the set of all functions whose degree is 0?? For example, what are the degrees of the inverse hyperbolic functions, the inverse trigonometric functions, and the erf?? Georgia guy (talk) 22:38, 11 January 2014 (UTC)[reply]

The article should really say asymptotically degree 0 since it only describes the behavior of the function as x→∞. I believe the usual way of stating the condition would be f=O(x^ϵ) for ϵ>0. I don't see how the question would make sense for functions whose domain does not include a neighborhood of ∞. The section is unsourced so it's hard to tell if that definition of "degree 0" is one that actually appears in the literature. --RDBury (talk) 04:08, 12 January 2014 (UTC)[reply]

According to the definition the set of all functions whose degree is 0 is

\{f:\deg(f)=0\}=\{f:\lim _{x\rightarrow \infty }{\frac {\log(f(x))}{\log(x)}}=0\}

. What more do you want? Bo Jacoby (talk) 22:39, 15 January 2014 (UTC).[reply]