[seqfan] Re: Close sequences again
Michael Porter
ic_designer at verizon.net
Tue Oct 20 21:04:00 CEST 2009
I was actually thinking of the first expression:
> (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic
> residues of prime(n)) / prime(n)
n=1
prime(n)=2
0 is a quadratic residue
1 is a quadratic residue
sum of non-residues=0 (A)
sum of residues=1 (B)
(A-B) / prime(n) = -1/2
n=2
prime(n)=3
0 is a quadratic residue
1 is a quadratic residue
2 is a quadratic non-residue
sum of non-residues=2 (A)
sum of residues=1 (B)
(A-B) / prime(n) = 1/3
- Michael
--- On Tue, 10/20/09, Christopher Gribble <chris.eveswell at virgin.net> wrote:
From: Christopher Gribble <chris.eveswell at virgin.net>
Subject: [seqfan] Re: Close sequences again
To: "'Sequence Fanatics Discussion list'" <seqfan at list.seqfan.eu>
Date: Tuesday, October 20, 2009, 2:50 AM
Michael,
Looks like my plain text table got garbled, so I'll try again:
n 1
prime(n) 2
floor(j^2/prime(n)), j=1 0
sum(floor(j^2/prime(n))), j=1:prime(n)-1) (A) 0
floor((prime(n)-2)(prime(n)-1)/3) (B) 0
A-B 0
n 2
prime(n) 3
floor(j^2/prime(n)), j=1 0
floor(j^2/prime(n)), j=2 1
sum(floor(j^2/prime(n))), j=1:prime(n)-1) (A) 1
floor((prime(n)-2)(prime(n)-1)/3) (B) 0
A-B 1
Chris
-----Original Message-----
From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
On Behalf Of Michael Porter
Sent: 20 October 2009 09:12
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Close sequences again
> (Sum of the quadratic non-residues of prime(n) - Sum of the quadratic
> residues of prime(n)) / prime(n). I discovered that this was equivalent
to
> (sum ( floor (j^2 / prime(n))), j = 1:prime(n) - 1) - floor ((prime(n) -
> 2)(prime(n) - 1) / 3) and that this could be extended to the naturals.
That seems like a non-trivial result, so I'm glad to see you have already
added it to OEIS (A165951). But for a(1) and a(2), prime(n)=2 and 3, the
expression evaluates to -1/2 and 1/3, right?
> A166128
> A166129
> A166130
> A166468
A166468 is the one we were discussing earlier, and if I read it correctly,
A166128 is just prime(n)-1 with a(1) set to 0, so a similar argument
applies. I can't seem to get to A166129 and A166130.
- Michael
_______________________________________________
Seqfan Mailing list - http://list.seqfan.eu/
_______________________________________________
Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list