# [seqfan] Re: Generally known (to those who know of such things)

Richard Mathar mathar at strw.leidenuniv.nl
Fri Nov 20 19:59:18 CET 2009

```Quoting http://list.seqfan.eu/pipermail/seqfan/2009-November/002990.html

> From seqfan-bounces at list.seqfan.eu Fri Nov 20 19:31:02 2009
> To: seqfan at list.seqfan.eu
> Date: Fri, 20 Nov 2009 13:04:34 -0500
> Subject: [seqfan] Re: Generally known (to those who know of such things)
>
> I don't understand, again, why the primes (p1) of form 4n+1 differ in
> behaviour from those (p3) of form 4n+3 in the following:
>
> u=Sum=1..p1;  k (p1-k) ) is strictly increasing in function of p1, while
> v=Sum(k=1..p3;  k (p3-k) ) is not (in function of p3).

So executing the sums gives
u = p1*(p1+1)*(p1-1)/6
v = p3*(p3+1)*(p3-1)/6
and one would expect all members of u and v to be in A127920.
I get the same sequence as Franklin, and the v sequence

>
> v is descending from v(37) to v(38)
> {4, 28, 66, 190, 322, 558, 946, 1316, 1888, 2278, 2982, 3476, 3652,
> 5768, 5992, 8636, 9170, 10008, 12382, 13366, 15698,
>  16826, 20628, 21492, 22788, 26314, 26786, 32026, 33132, 37872, 39566,
> 40752, 47892, 54114, 55608, 61766, 71082, 70464
>
> I checked 'u' way up to 10,000 and found no 'drops'.

remains obscure because the numbers do not match this criterion.

Generally speaking, the primes == -1 (mod 4) are just getting a slower start initially,
exhibited by A007351, A007350. A002144(36)=389 whereas A002145(36)=347.

RJM

```