A263920 - OEIS

%I #6 Feb 16 2025 08:33:27

%S 7,47,57,99,117

%N A positive integer n is in this sequence iff arctan(n)^2 can be represented as Sum_{0<k<n} c(k)*arctan(k)^2 with rational c(k). The terms are in increasing order.

%C The terms given are certainly in the sequence. Although I lack a rigorous proof that no intermediate terms were omitted, an extensive computer search gave no other candidates in between.

%C It is an open question if the sequence is infinite.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/InverseTangent.html">Inverse Tangent</a>.

%e 7 is in the sequence, because arctan(7)^2 = -5*arctan(1)^2 + (10/3)*arctan(2)^2 + (2/3)*arctan(3)^2.

%e 47 is in the sequence, 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.

%Y Cf. A005528, A002312.

%K nonn,hard,more,changed

%O 1,1

%A _Vladimir Reshetnikov_, Oct 29 2015