A230126 - OEIS

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

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OFFSET

0,2

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

Table of n, a(n) for n=0..16.

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

PROG

(PARI)

{

a=1; s=0; S=Pi/2;

while(1, s+=atan(1/a); if(s>S,

S+=Pi/2; print(a)); a++)

}

CROSSREFS

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