[seqfan] Re: Prime production length of n

Maximilian Hasler maximilian.hasler at gmail.com
Sat Mar 3 22:37:46 CET 2012

On Fri, Mar 2, 2012 at 4:59 PM, Neil Sloane <njasloane at gmail.com> wrote:
> http://m-hikari.com/ija/ija-2012/ija-1-4-2012/xushaojiIJA1-4-2012.pdf.
> seems to suggest that there are two sequences that should be in the OEIS:
> the prime production length of the polynomial x^2+x+n,
> where n runs through the positive (or negative) integers.

I started with

Prime production length of the polynomial P=x^2+x+prime(n): max { k>0
| P(x) is prime for all x=0,...,k-1 }

There are simple variations which essentially contain the same information:
One could also consider P=x^2-x+n,
which would yield zero for all composite numbers,
unless one would start with x=1,2,3,...

One could also consider Q=x^2 - x + n (or +prime(n)),
again with x=0,... or x=1,...
(This sign was Euler's choice).
It amounts to increase all values by 1
(since Q(x) = P(-x) = P(x-1).)

Following the previous remark, allowing negative x
also would not add more information.

Would it be worth to add the "aeration" with prime(n) replaced by n,
[not really aeration, esp. when the start x=0 is changed to x=1 in the "+" case]
and/or the version(s) with switched sign where all (nonzero) values
are augmented by 1 ?


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