cp's OEIS Frontend

A230126 Smallest value of k such that Sum_{j=1..k} arctan(1/j) > n*Pi/2.

%I A230126 #33 Mar 17 2021 02:54:57
%S A230126 1,4,17,82,396,1905,9165,44088,212082,1020218,4907734,23608545,
%T A230126 113568371,546318080,2628050766,12642178765,60814914995
%N A230126 Smallest value of k such that Sum_{j=1..k} arctan(1/j) > n*Pi/2.
%C A230126 Equivalently, integers k such that (1+i)*(2+i)*...*(k+i) is not in the same quadrant of the complex plane that (1+i)*(2+i)*...*(k-1+i) is in (if one of these numbers lies on the real or imaginary axis, it is taken to be in the quadrant immediately clockwise from it).
%C A230126 The only time that (1+i)*(2+i)*...*(k+i) lies on the real or imaginary axis is when k = 3, which follows from a result of Cilleruelo (see links). - _Nathaniel Johnston_, Dec 27 2013
%C A230126 The ratio between successive terms quickly approaches exp(Pi/2), which can be proved using the Taylor series of the arctangent function and the (basic) definition of Euler's constant.
%H A230126 Javier Cilleruelo, <a href="http://matematicas.uam.es/~franciscojavier.cilleruelo/Papers/squares-sinlogo.pdf">Squares in (1^2 + 1)*...*(n^2 + 1)</a>, Journal of Number Theory 128:8 (2008), pp. 2488-2491.
%o A230126 (PARI)
%o A230126 {
%o A230126 a=1;s=0;S=Pi/2;
%o A230126 while(1,s+=atan(1/a);if(s>S,
%o A230126 S+=Pi/2;print(a));a++)
%o A230126 }
%Y A230126 Cf. A042972 (exp(Pi/2)), A231530, A231531.
%K A230126 nonn,more
%O A230126 0,2
%A A230126 _James G. Merickel_, Oct 10 2013
%E A230126 a(17) added by _James G. Merickel_, Oct 14 2013