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

1, 4, 17, 82, 396, 1905, 9165, 44088, 212082, 1020218, 4907734, 23608545, 113568371, 546318080, 2628050766, 12642178765, 60814914995

Offset: 0

Author

James G. Merickel, Oct 10 2013

Keywords

• nonn
• more

Comments

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).

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

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.

Links

• Javier Cilleruelo, Squares in (1^2 + 1)*...*(n^2 + 1), Journal of Number Theory 128:8 (2008), pp. 2488-2491.

Crossrefs

Cf. A042972 (exp(Pi/2)), A231530, A231531.

Programs

PARI

{
a=1;s=0;S=Pi/2;
while(1,s+=atan(1/a);if(s>S,
S+=Pi/2;print(a));a++)
}

Extensions

a(17) added by James G. Merickel, Oct 14 2013