[math-fun] Flags and antihistamines

[David Wilson]

I don't know where to point you, but this subject has been treated before. 
The last time we dealt with them, we were calling them "flag products".

There are two (interesting) kinds of flag products, the outer flag product, 
where the longer row/column is at the edge, and the inner flag product, 
where the shorter row/column is at the edge.

The star pattern on the US flag is an example of an outer flag product. It 
is 5 stars high and 6 stars wide at the edge and has 50 stars.  Letting "o" 
be the outer product, we have

   5 o 6 = 50.

It turns out that the outer flag product "o" is conjugate to the standard 
integer product "*" via the pretty identity

  a o b = c   <=>   (2a-1) * (2b-1) = (2c-1).

Thus 5 o 6 = 50 corresponds to 9 * 11 = 99.

Similarly, the inner flag product is like your pill card, which has 1 pill 
on one edge, 3 pills on the other and 10 pills altogether.  Letting "i" be 
the inner flag product, we have

   1 i 3 = 10.

The inner flag product "i" also has a pretty conjugacy with the standard 
integer product "*":

  a i b = c  <=>   (2a+1) * (2b+1) = (2c+1).

Again, 1 i 3 = 10 corresponds to 3 * 7 = 21.

Now, an outer flag composite would be a number of the form c = a o b where a 
 >= 2 and b >= 2, that is, the number of stars in a US flag pattern 2 or more 
stars on each edge.  This corresponds to (2c-1) = (2a-1) * (2b-1), that is 
the odd number 2c-1 is a product of two odd numbers (2a-1) >= 3 and (2b-1) 
 >= 3.  This is true precisely when 2c-1 is composite.  So an outer flag 
composite is a number c where 2c-1 is a standard composite.

Similarly, an inner flag composite is a number of the form c = a i b where a 
 >= 1 and b >= 1 (why 1 and not 2?).  This corresponds (2c+1) = (2a+1) * 
(2b+1), and by similar reasoning to the last paragraph, an inner flag 
composite is a number c where 2c+1 is a composite.

Your sequence 1, 2, 3, 6, 9, 15... are precisely the numbers c which are 
neither an inner flag composite nor an outer flag composite.  These would be 
the numbers such that neither 2c-1 nor 2c+1 is a standard composite, that 
is, 2c-1 and 2c+1 are both either 1 or prime.  These include c = 1 together 
with those c such that 2c-1 and 2c+1 are a twin prime pair.  So your 
sequence is A040040 prepended with 1.

And yes, all numbers starting at 3 are divisible by 3.

----- Original Message ----- 
From: "Kerry Mitchell" <lkmitch at att.net>
To: "math-fun" <math-fun at mailman.xmission.com>
Sent: Monday, October 17, 2005 8:07 PM
Subject: [math-fun] Flags and antihistamines


> The flag of the United States has 50 stars arranged in an alternating 
> pattern:  a row of six stars, then a row of five, etc.  Alternatively, the 
> star field can be viewed as a column of five stars, then a column of four, 
> etc.  This assemblage can also be seen as a four-by-five grid of stars 
> interlaced within a five-by-six grid.  See 
> http://en.wikipedia.org/wiki/Image:Flag_of_the_United_States.svg for a 
> good illustration.
>
> Since it is becoming allergy season for me, I happened to whip out my 
> antihistamines and noticed that the 10 pills on a card are arranged in 
> three rows:  a row of three, a row of four, and a row of three.  Or, in a 
> one-by-four grid within a two-by-three grid.
>
> This got me wondering:  what numbers can be represented as the total 
> number of nodes (or stars or pills) where the nodes are laid out in two 
> interlaced rectangular grids?  For this problem, I’m only considering the 
> case where the numbers of rows for the two grids differ by exactly one, as 
> do the numbers of columns (so a 1x4 in a 2x3 is ok, but not a 1x4 in a 
> 3x6).  This constraint makes it easy to alternate rows and columns to make 
> adjacent nodes align diagonally instead of vertically.
>
> What I found more interesting than the numbers that can be represented is 
> the set of numbers that cannot be represented:  1, 2, 3, 6, 9, 15, etc. 
> (not in OEIS).  1 and 2 are trivial because there must be at least three 
> rows.  For every element after 2 (that I’ve found) is a multiple of 3. 
> The elements exhibit a decidedly non-regular pattern, and occur about half 
> as frequently as primes.  Can anyone point me to more information about 
> this before I foolishly think that discovered something?
>
> Kerry Mitchell
> --
> lkmitch at att.net
> www.fractalus.com/kerry
>
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