[seqfan] Re: A sequence inspired by Størmer numbers (A005528).

[Roland Bacher]

Dear seqfans,

There is for example also the relation

41647*a(3)+17340*a(4)-15115*a(5)-36283*a(7)-28865*a(8)-5780*a(13)
+385*a(18)+14450*a(21)-10115*a(38)+14450*a(47)+14695*a(57)

and the relation

-138456*a(4)-25570*a(5)+.... +41647*a(99)

where a(n)=arctan(n)^2

An (incomplete and very sketchy) explanation for the origin and
existence of such relations is as follows:

The formula:

arctan(x)=(log(1+ix)-log(1-ix))/(2i)

shows that we have to consider prime factors
of 1+n^2 for natural integers n
(such factors are of course exactly 2 and
all primes congruent to 1 modulo 4).

Consider variables representing logarithms 
of the underlying complex primes
in Gaussian integers. (one needs two variables 
$x_p,y_p$ for every real prime).

Linear combinations of arctan(n)^2
can then be interpreted as 
quadratic forms (over the gaussian integers
without error) in these variables 
which simplify to zero.

The question of an infinity of such relations
can probably be adressed arithmetically
(I guess the answer is yes).

There are less relations than among 
linear combinations of arctangents
(for linear relations among arctan(n)
you get an additional relation every
time all prime-factors of 1+n^2
occured already previously and the coefficients
can be deduced from prime-factorizations of 1+j^2)
since we have to deal with complex coefficients
(and there is perhaps also
pi coming in, I did not think it through 
completely).

>From a computational point of view,
I generated all integers $n$ up to some 
bound with 1+n^2 involving only small 
primes.

One searches then for a relation
using for example the PSLQ algorithm 
under maple (I did it using 1500
Digits of precision).

Pushing the study a little bit further,
It should be possible to write down 
everything theoretically using 
factorisation in Gaussian integers.

Best wishes,  Roland Bacher



On Wed, Oct 28, 2015 at 10:22:49PM -0700, Vladimir Reshetnikov wrote:
> Dear SeqFans,
> 
> Experimenting with linear combinations of arctangents of integers, I
> discovered a sequence that turned out to be already known and
> studied: Størmer numbers http://oeis.org/A005528.
> 
> It inspired me to ask a similar question about squares of arctangents.
> Consider an increasing sequence A of positive integers, such that a
> positive integer n is in A, iff the expression arctan(n)^2 can be
> represented as a linear combination of terms arctan(k)^2 with rational
> coefficients, where k are positive integers less than n.
> 
> For example, n = 7 is in A, because
> arctan(7)^2 = -5*arctan(1)^2 + 10/3*arctan(2)^2 + 2/3*arctan(3)^2.
> 
> Likewise, n = 47 is also in A, because
> arctan(47)^2 = 2939/210*arctan(2)^2 - 125/21*arctan(3)^2 - 6/5*arctan(4)^2
> - 12/7*arctan(5)^2 - 29/7*arctan(7)^2 + 15/7*arctan(8)^2 + 2/5*arctan(13)^2
> + 11/7*arctan(18)^2 - arctan(21)^2 + 7/10*arctan(38)^2.
> 
> Can you suggest an efficient way to compute the sequence A? Is it infinite?
> 
> Also, we can consider a sequence whose n-th term is a number of such
> representations for arctan(n)^2. So, indices of its non-zero terms
> constitute the sequence A.
> 
> Any ideas are appreciated.
> 
> --
> Thanks
> Vladimir
> 
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