[seqfan] Re: : The alternating sum in A196020

[Charles Greathouse]

Conjecture checks to 10^6. Here's the simple code I'm using for the
alternating sum:

alt(n)=my(s=2*n-1,t);for(k=2,sqrtint(2*n),t=n-k*(k+1)/2;if(t>=0&&t%k==0,t=t/k*2+1;s-=(-1)^k*t));s

Note the congruence condition n = k(k+1)/2 mod k.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Sun, Nov 17, 2013 at 11:57 AM, William Keith <william.keith at gmail.com>wrote:

> Seems plausible to me.  I would suggest something along the following
> lines: you are including or excluding divisors of size k by adding the k-th
> column.  The k-th column becomes "necessary" when you must add or cancel
> divisors of sizes 1, 2, ... and k, when you have only had to add or cancel
> at most k-1 of them before, ergo it appears at n=k(k+1)/2.
>
> (Conjecture checks out to n=100 000 easily, btw.)
>
> William Keith
>
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