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%I #3 Mar 30 2012 16:50:12
%S 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,
%T 12,12,12,12,12,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,29,
%U 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
%N Smallest integer q >= 1 such that difference between q*sqrt(2) and the nearest integer is <= 1/n.
%C 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,...
%D J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge, 1957.
%p Digits:=200; M1:=200; th:=x->abs(x-round(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)));
%Y Cf. Pell numbers A000129; A108688, A108689.
%K nonn
%O 2,2
%A _N. J. A. Sloane_, Jun 18 2005
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