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This page contains notes about the status of the OEIS wiki. It provides a brief description of the wiki setup. It will also be used to announce changes in the structure of the wiki, so it may be useful to add it to your watchlist by clicking on the watch tab at the top of the page.

Users can create and modify discussion pages here. The discussion pages are a reasonable place to make comments about the structure of the wiki.

Do not use the discussion pages to suggest changes to OEIS sequence entries.

An account is necessary to contribute to the Wiki. To request an account, click here. There was a time when the username was required to be your Real Name. This is no longer required, but is strongly recommended. If this evolving statement causes you to want to change your username, send email to David, and I can make the change.

A few preliminary pages (most of these need to be updated!)

[talk]

Sequence of the Day for April 19

A092287: \prod_{j = 1}^n \prod_{k = 1}^n \gcd(j, k),\ n \ge 0.

{ 1, 1, 2, 6, 96, 480, 414720, 2903040, ... }

Peter Bala conjectures that the order of the primes in the prime factorization of a(n) is given by the formula

 \operatorname{ord}_p\ a(n) = \sum_{k = 1}^{\lfloor \log_p(n) \rfloor} \left\lfloor\frac{n}{p^k}\right\rfloor ^2 = \left\lfloor \frac{n}{p} \right\rfloor ^2 + \left\lfloor \frac{n}{p^2} \right\rfloor ^2 + \left\lfloor \frac{n}{p^3} \right\rfloor ^2 + \cdots .

Charles R Greathouse IV proved Bala's conjecture very recently.

Comparing this with the de Polignac–Legendre formula for the prime factorization of n!, i.e.

 \operatorname{ord}_p\ n! = \sum_{k = 1}^{\lfloor \log_p(n) \rfloor} \left\lfloor\frac{n}{p^k}\right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \cdots ,

this suggests that a(n) can be considered as a generalization of the factorial numbers (the product between braces is obviously 1 if n is noncomposite)

\frac{a(n)}{n!} = \left( \prod_{k = 1}^{n-1} \gcd(n, k) \right)^2 \frac{a(n-1)}{(n-1)!},\quad n \ge 1.

Recurrence:

a(0) := 1;\ a(n) := n \left( \prod_{k = 1}^{n-1} \gcd(n, k) \right)^2 a(n-1),\quad n \ge 1.

Formula:

a(n) = n! \left( \prod_{j = 1}^{n} \prod_{k = 1}^{j-1} \gcd(j, k) \right) ^2,\quad n \ge 0.

See GCD matrix generalization of the factorial.

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