Re: [math-fun] Does this int seq have a name?
Yes, the 2nd sequence consists of the numerators to the convergents of 2pi. The 1st sequence is similar to the second, except that these are numerators of convergents that are always slightly > 2pi. More precisely, the 2nd sequence are N>1 s.t. 0<mod(N,2pi)<mod(M,2pi) for 0<M<N. The 1st sequence: N>1 s.t. 0<|rem(N,2pi)|<|rem(M,2pi)| for 0<M<N; rem(x,y) is least absolute remainder of x/y. Here's the setup (see yesterday's post re "exploring the circle group"): We all know that |exp(iN)|=1. But we're now interested in those N for which exp(iN)~1 (i.e., w/o the absolute value). exp(iN)=exp(i)^N "explores" the circle group uniformly; it just doesn't do it in a continuous fashion (unless someone here has a better ordering/metric for the circle group). Which now raises an interesting question: Given a unit complex number z (i.e., |z|=1), we'd like to find a good N, s.t. exp(i)^N ~ z. I.e., given a real tolerance epsilon>0 and complex z s.t. |z|=1, we'd like to find the smallest N>0, s.t. |exp(i)^N - z|<epsilon. Are there efficient algorithms to find such N's ? This question is almost, but not quite, the same as discreet logs (i.e., logs capable of protecting privacy). Also, check out CORDIC: https://en.wikipedia.org/wiki/CORDIC At 10:12 AM 3/16/2018, Allan Wechsler wrote:
OEIS recognized the second one: see http://oeis.org/A046955
You should submit the first, and add cross-references.
On Fri, Mar 16, 2018 at 12:27 PM, Henry Baker <hbaker1@pipeline.com> wrote:
1,7,13,19,44,377,710,104703,...
and another closely related sequence
1,6,19,25,44,333,710,103993,...
They are both monotonically increasing.
This posting should have been made on pi day.
The sequences are related to CORDIC.
I agree with Allan - please add the first sequence to the OEIS, with a cross-ref to the second Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Mar 16, 2018 at 2:13 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Yes, the 2nd sequence consists of the numerators to the convergents of 2pi.
The 1st sequence is similar to the second, except that these are numerators of convergents that are always slightly > 2pi.
More precisely, the 2nd sequence are N>1 s.t. 0<mod(N,2pi)<mod(M,2pi) for 0<M<N.
The 1st sequence: N>1 s.t. 0<|rem(N,2pi)|<|rem(M,2pi)| for 0<M<N; rem(x,y) is least absolute remainder of x/y.
Here's the setup (see yesterday's post re "exploring the circle group"):
We all know that |exp(iN)|=1.
But we're now interested in those N for which exp(iN)~1 (i.e., w/o the absolute value).
exp(iN)=exp(i)^N "explores" the circle group uniformly; it just doesn't do it in a continuous fashion (unless someone here has a better ordering/metric for the circle group).
Which now raises an interesting question:
Given a unit complex number z (i.e., |z|=1), we'd like to find a good N, s.t. exp(i)^N ~ z.
I.e., given a real tolerance epsilon>0 and complex z s.t. |z|=1, we'd like to find the smallest N>0, s.t.
|exp(i)^N - z|<epsilon.
Are there efficient algorithms to find such N's ?
This question is almost, but not quite, the same as discreet logs (i.e., logs capable of protecting privacy).
Also, check out CORDIC:
https://en.wikipedia.org/wiki/CORDIC
At 10:12 AM 3/16/2018, Allan Wechsler wrote:
OEIS recognized the second one: see http://oeis.org/A046955
You should submit the first, and add cross-references.
On Fri, Mar 16, 2018 at 12:27 PM, Henry Baker <hbaker1@pipeline.com> wrote:
1,7,13,19,44,377,710,104703,...
and another closely related sequence
1,6,19,25,44,333,710,103993,...
They are both monotonically increasing.
This posting should have been made on pi day.
The sequences are related to CORDIC.
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