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A230126
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Smallest value of k such that Sum_{j=1..k} arctan(1/j) > n*Pi/2.
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0
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1, 4, 17, 82, 396, 1905, 9165, 44088, 212082, 1020218, 4907734, 23608545, 113568371, 546318080, 2628050766, 12642178765, 60814914995
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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PROG
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(PARI)
{
a=1; s=0; S=Pi/2;
while(1, s+=atan(1/a); if(s>S,
S+=Pi/2; print(a)); a++)
}
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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