# [seqfan] Re: A172043 and many more

Joerg Arndt arndt at jjj.de
Wed Mar 3 09:30:43 CET 2010

```Dear Vincenzo,
please edit (or remove) the sequence A172043.

* Klaus Brockhaus <klaus-brockhaus at t-online.de> [Mar 03. 2010 09:11]:
> %e A172193 For n=0, a(0)=1; n=1, a(1)=37; n=2, a(2)=83; n=3, a(3)=139

I cannot make any sense of the title.

Note that (n^2+1) factors into (2 and) primes
of the form 4k+1 for all n

The comment " See sequence of prime:(37, 83, 139, 281, 367,..."
makes no sense at all.

? v=[1, 37, 83, 139, 205, 281, 367, 463, 569, 685, 811, 947, 1093, 1249, 1415];
? for(j=1,#v,print([j,factor(j^2+1)]))
[1, Mat([2, 1])]
[2, Mat([5, 1])]
[3, [2, 1; 5, 1]]
[4, Mat([17, 1])]
[5, [2, 1; 13, 1]]
[6, Mat([37, 1])]
[7, [2, 1; 5, 2]]
[8, [5, 1; 13, 1]]
[9, [2, 1; 41, 1]]
[10, Mat([101, 1])]
[11, [2, 1; 61, 1]]
[12, [5, 1; 29, 1]]
[13, [2, 1; 5, 1; 17, 1]]
[14, Mat([197, 1])]
[15, [2, 1; 113, 1]]

The sequence appears to be simply 5*n^2+31*n+1 (?)
? vector(15,n,5*n^2+31*n+1)
[37, 83, 139, 205, 281, 367, 463, 569, 685, 811, 947, 1093, 1249, 1415, 1591]

What is the significance of this?

Btw. the generating function can (for seqs like this
always be given):
? ggf(vector(15,n,5*n^2+31*n+1))
(x^2 - 28*x + 37)/(-x^3 + 3*x^2 - 3*x + 1)
\\ == (x^2 - 28*x + 37)/(x-1)^3

cheers,   jj

```