

A105960


Smallest integer q >= 1 such that difference between q*sqrt(2) and the nearest integer is <= 1/n.


2



1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29
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OFFSET

2,2


COMMENTS

Theorem 1 in Cassels says given real numbers x and Q>1, there is an integer q such that 0 < q < Q and the difference between qx and the nearest integer is <= 1/Q. This sequence arises from taking x = sqrt(2) and Q = n = 2,3,4,...


REFERENCES

J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge, 1957.


LINKS

Table of n, a(n) for n=2..77.


MAPLE

Digits:=200; M1:=200; th:=x>abs(xround(x)); f:=proc(x) local Q, q, t1, x1; t1:=[]; for Q from 2 to M1 do x1:=evalf(1/Q); q:=1; while th(q*x) > x1 do q:=q+1; od; t1:=[op(t1), q]; od; t1; end; f(evalf(sqrt(2)));


CROSSREFS

Cf. Pell numbers A000129; A108688, A108689.
Sequence in context: A316660 A098101 A257670 * A081290 A168256 A123081
Adjacent sequences: A105957 A105958 A105959 * A105961 A105962 A105963


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jun 18 2005


STATUS

approved



