This site is supported by donations to The OEIS Foundation.

# User talk:Matthew Vandermast

## A212172

Name: Row n of table gives second signature of n (nonincreasing version): list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 when no such exponent exists.

You wrote:

Each sequence of integers with a given second signature (S) has a positive density, the highest of these being 6/Pi^2 = .607927... for A005117 ((S) = (0)).

True for S = 0: 6/Pi^2 = .607927... (squarefree numbers) (see also squareful numbers and squarefull numbers)

Not true for S = {2}: square of a prime, since primes have asymptotic density of 0 and squares have asymptotic density of 0.

Not true for S = {k, k ≥ 2}: k th power of a prime, since (...) and k th powers have asymptotic density of 0.

Daniel Forgues 02:27, 11 June 2012 (UTC)

• 1. Thanks for your interest, Daniel.
• 2. An integer with second signature (2) can have any number of 1s as exponents in its prime factorization. The 1s just aren't counted in the second signature.
• In other words, the sequence of integers with second signature (2) (see https://oeis.org/A060687) contains more than just squares of primes. It includes every number of the form n*p^2, where n is squarefree and coprime to p (and p of course represents a prime). I believe its positive density is the sum of the reciprocals of A036690 (https://oeis.org/A036690) multiplied by 6/Pi^2. This product seems to be about .2015... (but don't consider this too accurate).
• 3. Maybe you'd be interested in finding more digits of this density and the other densities?
• 4. I'm not yet very adept at editing this Wiki (to put it mildly). Let's see how this goes. Matthew Vandermast 18:47, 11 June 2012 (UTC)

Thanks for your reply! I had a vague feeling that I might be wrong in my reasoning, but couldn't see where. Is there a proof (or is it a conjecture) for the asymptotic densities? — Daniel Forgues 03:59, 12 June 2012 (UTC)
Sorry for the delay, Daniel. I wrote a fairly long reply two days ago, but forgot to save the changes! But - yes, there's definitely no sequence of numbers of a given second signature with asympotic density 0.
In fact, there's no sequence of numbers with any given largest powerful divisor (cf. A057521) whose asymptotic density is 0. And the numbers whose largest powerful divisors have prime signature {S} are exactly the numbers with second signature {S}.
I imagine you know this, but perhaps someone may read this who doesn't: The density of squarefree numbers (6/Pi^2, or .607927...) equals
Product_ (p=1 to infinity) (p^2-1)/(p^2)
An obvious but key fact: It's a product of infinitely many positive fractions. To find the density of the sequence of numbers whose largest powerful divisor = n, you replace a finite number of those fractions with the same finite number of smaller positive fractions. And, of course, the new product is still positive. Examples:
n=4: replace (3/4) by (1/8)
(Why? Because 3/4 of integers have a 2-exponent of 0 or 1 in their prime factorizations; 1/8 have a 2-exponent of 2)
new product: 1/Pi^2 = .101321...
All of these numbers have second signature {2}, so this is enough to prove those numbers have a positive density. We also have a second signature of {2} when n = 9 (replace 8/9 by 2/27); 25 (replace 24/25 by 4/125); or any number p^2 (replace (p^2-1)/(p^2) by (p-1)/(p^3)).
n=72: replace (3/4 * 8/9) by (1/16 * 2/27)
(The reasoning is analogous to the reasoning for 4)
new product = 1/36*Pi^2 = .002814...
The numbers with second signature {3,2} include not only these numbers, but all other numbers whose powerful divisor has a prime signature of {3,2}.
Clearly, this works for numbers of any second signature. So the numbers of every second signature have a positive density.

A variant name suggestion for A212172:

Concatenation of second signatures of n (nonincreasing version), where second signature of n is list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 when no such exponent exists.

Daniel Forgues 02:35, 11 June 2012 (UTC)

• Thanks, but I prefer "table" to "concatenation." Matthew Vandermast 18:47, 11 June 2012 (UTC)

To avoid using 0 as a kludge for squarefree numbers, I would suggest that you submit the sequence

Concatenation of second signatures of squareful numbers (nonincreasing version), where second signature of n is list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 when no such exponent exists.

That way, you just skip over all those squarefree numbers. — Daniel Forgues 02:50, 11 June 2012 (UTC)

• I will be submitting both sequences, and I hope this increases the chance that people will find at least one of them. Matthew Vandermast 18:47, 11 June 2012 (UTC)

By the way, a multiset is also called a bag... — Daniel Forgues 07:30, 12 June 2012 (UTC)

This is a new section of my talk page!

```Let's see how it looks and exactly where it appears.
```

Here's some wikitext and HTML tips:
If the first character of a line is a space, the text is in fixed-font (instead of proportional font) and appears in a box:
```''Text'' yields italic Text.
```
''Text'' yields italic Text.
'''Text''' yields bolded Text.
'''''Text''''' yields bolded italic Text.
To show text in fixed-font, you may use <tt> ... </tt>. To show code, you may use <code> ... <code> or <pre> ... </pre>.

## Start a of new section

You start a new section with == (at start of line and after section title), and different levels of subsections with ===, ====, =====, ====== (last level of subsections with Mediawiki). — Daniel Forgues 21:36, 12 June 2012 (UTC)

Thanks for the tips, Daniel! May I show off my new skills (:-)) to ask a question?

## How can I start a Wiki discussion page on a particular mathematical topic?

Someone suggested I do this the other day, but I have no idea how, and my search for instructions wasn't successful. Thanks to Daniel or anyone else who can help.

(Thanks again, Daniel. Dang, that subject heading looks good. :))

When you search for a page that doesn't exist (e.g. Lucabonacci numbers), the link is in red, just click on it to create the page. The talk page Talk:Lucabonacci numbers should only be created after the article page Lucabonacci numbers exists. There are some pages of general interest for OEIS Wiki, e.g. Features Wishlist where the article page itself is sort of a hub to exchange and discuss ideas about a topic such as a wishlist of features for the software for the main (curated) OEIS, but these are the exception. — Daniel Forgues 04:22, 15 June 2012 (UTC)

Thanks! Matthew Vandermast 19:23, 15 June 2012 (UTC)