Template:Sequence of the Day for April 19

Intended for: April 19, 2014

Timetable

• First draft entered by Peter Luschny on April 1, 2013 ✓
• Draft reviewed by Alonso del Arte on April 2, 2013 ✓
• Draft approved by Daniel Forgues on April 10, 2013 ✓

Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.

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A092287:

∏ n  j  = 1 ∏ n k  = 1 gcd (  j, k ), n   ≥   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

${\begin{array}{l}\displaystyle {\operatorname {ord} _{p}\ a(n)=\sum _{k=1}^{\lfloor \log _{p}(n)\rfloor }\left\lfloor {\frac {n}{p^{k}}}\right\rfloor ^{2}=\left\lfloor {\frac {n}{p}}\right\rfloor ^{2}+\left\lfloor {\frac {n}{p^{2}}}\right\rfloor ^{2}+\left\lfloor {\frac {n}{p^{3}}}\right\rfloor ^{2}+\cdots .}\end{array}}$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.

${\begin{array}{l}\displaystyle {\operatorname {ord} _{p}\ n!=\sum _{k=1}^{\lfloor \log _{p}(n)\rfloor }\left\lfloor {\frac {n}{p^{k}}}\right\rfloor =\left\lfloor {\frac {n}{p}}\right\rfloor +\left\lfloor {\frac {n}{p^{2}}}\right\rfloor +\left\lfloor {\frac {n}{p^{3}}}\right\rfloor +\cdots ,}\end{array}}$

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)

${\begin{array}{l}\displaystyle {{\frac {a(n)}{n!}}=\left(\,\prod _{k=1}^{n-1}\gcd(n,k)\right)^{2}{\frac {a(n-1)}{(n-1)!}},\quad n\geq 1.}\end{array}}$

Recurrence:

${\begin{array}{l}\displaystyle {a(0):=1;\ a(n):=n\left(\,\prod _{k=1}^{n-1}\gcd(n,k)\right)^{2}a(n-1),\quad n\geq 1.}\end{array}}$

Formula:

${\begin{array}{l}\displaystyle {a(n)=n!\left(\,\prod _{j=1}^{n}\prod _{k=1}^{j-1}\gcd(j,k)\right)^{2},\quad n\geq 0.}\end{array}}$

A??????

(   ∏ n  j  = 1 ∏  j  − 1 k  = 1 gcd (  j, k )) 2, n   ≥   0

.

{1, 1, 1, 1, 4, 4, 576, 576, 147456, 11943936, 76441190400, 76441190400,...}

A224479

∏ n  j  = 1 ∏  j  − 1 k  = 1 gcd (  j, k ), n   ≥   0

.

{1, 1, 1, 1, 2, 2, 24, 24, 384, 3456, 276480, 276480, 955514880, 955514880, ...}

​-------------------------

* See GCD matrix generalization of the factorial.