Re: Some list questions

• To: mathgroup at smc.vnet.net
• Subject: [mg37140] Re: [mg37132] Some list questions
• From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
• Date: Sun, 13 Oct 2002 05:56:19 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Of course one can use standard "programming" techniques to answer this
and it will in fact be the most efficient method. But as you will
probably get lots of answers of this kind, I will do it in another way:
by exploiting a few standard built-in number theoretic functions which
are very closely connected with your problems.

Question 1:

In[1]:=
funct1[l_List]:=Outer[Times,Sequence@@(Divisors/@Power@@@l)]//Flatten

In[2]:=
funct1[{{2,3},{3,1},{5,1}}]

Out[2]=
{1,5,3,15,2,10,6,30,4,20,12,60,8,40,24,120}

Note what we did. We first converted your pairs {a,b}  back into powers
a^b then found all the divisors using the built in Divisors function,
then found all the products using Outer.

Question 2.

In[3]:=
minint[list1_,list2_]:=GCD[
Times@@Power@@@list1,Times@@Power@@@list2]//FactorInteger

In[4]:=
maxint[list1list1_,list2_]:=LCM[Times@@Power@@@list1,Times@@Power@@@list
2]\
//FactorInteger

e.g.

In[5]:=
list1 = {{2, 3}, {3, 4}, {5, 6}, {7, 2}, {17, 5}};

In[6]:=
list2 = {{2, 5}, {3, 2}, {5, 1}, {7, 3}};

In[7]:=
minint[list1,list2]

Out[7]=
{{2,3},{3,2},{5,1},{7,2}}

In[8]:=
maxint[list1,list2]

Out[8]=
{{2,5},{3,4},{5,6},{7,3},{17,5}}

Basically all we did was to use the built in functions GCD and LCM
after converting your lists of powers to numbers. Then we factored them
again.

In this case to finally factor an integer, which guarantees the
programs to be inefficient for large numbers. However if your original
list of pairs were indeed the result of using FactorInteger,  then you
should of course use versions of the above programs that can be applied
to the original un-factored integers. Indeed, in that case this is the
only efficient way to proceed.

Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/

On Saturday, October 12, 2002, at 06:05 PM, flip wrote:

> Hello (sorry for the long post),
>
> I have two seperate list questions that I was hoping to get help with.
>
> Question 1.
>
>  I have a variable length list similar to that generated by
> FactorInteger,
> that is {number, exponent} pairs.  An example follows.
>
> lista =  {{2,3},{3,1},{5,1}}  ... this is the number 2^3 * 3^1 * 5^1
>
> I want to generate a list of "all" the products of numbers from this
> list.
>
> I can tell that I get a total (3+1)*(1+1)*(1+1) = 4*2*2 = 16, products
> and I
> want a list showing all of those.
>
> These would be:
>
> 2^3 can generate {2^0, 2^1, 2^2, 2^3} = {1, 2, 4 ,8}
> 3^1 can generate {3^0,  3^1} = {3} ... we dont care about the
> duplicate "1"
> 5^1 can generate {5^0, 5^1} = {5} ... we dont care about the duplicate
> "1"
>
> Hence the 4*2*2 = 16 (the product of one more of the exponents) above.
>
> Next we should get 16 products (from these lists), namely (I left them
> as
> products below to show what I am after):
>
> {1,  2,  4,  8,  1*3,  2*3,  4*3,  8*3, 1*5, 2 * 5, 4* 5, 8* 5, 1*3*5,
> 2*3*5,  4*3*5,  8*3*5}
>
> If the list were lista = {{2,4}, {3,2}, {5, 3},{7^5}}, we would have
> (4+1)(2+1)(3+1)(5+1) = 360 products, for example and the return values
> should be a single list showing all of those.
>
> Question 2.
>
> I have two lists and want to generate two new lists from them.  These
> two
> lists are {number, exponent} pairs.
>
> In the first list, I want the "minimum intersection" of {number,
> exponent}
> pairs.
>
> In the second list, I want the "maximum union" of {number, exponent}
> pairs.
>
> Let me show an example:
>
> Input:
>
> list1 = {{2, 3}, {3, 4}, {5, 6}, {7, 2}, {17, 5}}
>
> list2 = {{2, 5}, {3, 2}, {5, 1}, {7, 3}}
>
> Output:
>
> minint = {{2, 3}, {3, 2}, {5, 1}, {7, 2}}
>
> Note: In this example we only kept those pairs where the intersection
> of the
> number exists and also keep the min power of those.
>
> maxint =  {{2, 5}, {3, 4}, {5, 6}, {7, 3}, {17, 5}}
>
> Note: In this example we kept the union of lists and also keep the max
> power
> of each.
>
> Thank you so much to this newsgroup for being so helpful.
>
> Flip
>
> Note: remove the "_alpha" from the email given to email me.
>
>
>
>
>
>

```

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