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

Vladimir Reshetnikov v.reshetnikov at gmail.com
Thu Oct 29 06:22:49 CET 2015


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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