A105960 Smallest integer q >= 1 such that difference between q*sqrt(2) and the nearest integer is <= 1/n.

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

Offset: 2

Author

N. J. A. Sloane, Jun 18 2005

Keywords

• nonn

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.

Crossrefs

Cf. Pell numbers A000129; A108688, A108689.

Programs

Maple

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)));