[seqfan] Re: A192787 (Re: Sequence relating to the Erdős–Straus conjecture)

Allan Wechsler acwacw at gmail.com
Tue Feb 19 23:21:06 CET 2013


As I said in the original thread, it would probably be worthwhile reverse
engineering exactly what the PARI code is calculating; it's some sequence,
even if it's not the advertised one.

On Tue, Feb 19, 2013 at 5:18 PM, Robert Munafo <mrob27 at gmail.com> wrote:

> I found PARI documentation and added print statements to the PARI code:
>
> ?
> a(n)=my(t=4/n,t1,s,c);for(a=1\t+1,3\t,t1=t-1/a;for(b=1\t1+1,2\t1,c=1/(t1-1/b);if(denominator(c)==1&&c>=b,s++;print("1/",a,"
> + 1/",b," + 1/",c))));s
>
> ? a(1)
> %2 = 0
> ? a(2)
> 1/1 + 1/2 + 1/2
> %3 = 1
> ? a(3)
> 1/1 + 1/4 + 1/12
> 1/1 + 1/6 + 1/6
> 1/2 + 1/2 + 1/3
> %4 = 3
> ? a(4)
> 1/2 + 1/3 + 1/6
> 1/2 + 1/4 + 1/4
> 1/3 + 1/2 + 1/6
> 1/3 + 1/3 + 1/3
> %5 = 4
> ? a(5)
> 1/2 + 1/4 + 1/20
> 1/2 + 1/5 + 1/10
> %6 = 2
>
>
> So the PARI code is wrong. It's counting "1/2 + 1/3 + 1/6" and "1/3 + 1/2
> + 1/6" as distinct.
>
>
>  From: Allan Wechsler
>>  Date: Tue, Feb 19, 2013 at 4:36 PM
>>  Subject: [math-fun] Sequence relating to the Erdős–Straus conjecture
>>  To: math-fun <math-fun at mailman.xmission.com>
>>
>> Let A(n) be the number of ways of expressing 4/n as the sum of three
>> integer reciprocals, where the mere permutation of a sum is regarded as
>> not
>> making a difference.
>>
>> Plainly 4/1 = 4 cannot be expressed as the sum of three reciprocals, so
>> A(1) = 0.
>>
>> 4/2 = 2 = 1/1 + 1/2 + 1/2, and there are no other solutions, so A(2) = 1.
>>
>> [...] Of course I wanted to know if A was in OEIS.  I calculated a few
>> more
>>
>>
>> Then I searched for "Straus", and quickly found A192787, which claims to
>> be
>> my A.  The trouble is, A192787(4) = 4, and I say A(4) = 3.
>>
>> Bear with me while I list my solutions, and then somebody tell me what I
>> missed.
>>
>> 4/4 = 1, so the problem is to partition 1 into three reciprocals.  I have
>> the following solutions:
>>
>> 1/2 + 1/3 + 1/6
>> 1/2 + 1/4 + 1/4
>> 1/3 + 1/3 + 1/3
>>
>> A192787 seems to be claiming that I missed one.  Charles R. Greathouse IV
>> was the sequence author, and I think he's a funster, so, Charles, if
>> you're
>> listening, can you tell me the missing dissection?
>>
>
> --
>   Robert Munafo  --  mrob.com
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>


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