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Numbers k which cannot be expressed as the sum of two positive integers k = a + b with a < b so that the continued fraction for a/b has all its partial quotients equal to 1 or 2.
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%I #8 Dec 30 2021 14:25:16

%S 1,2,4,23,37,53,59,61,83,103,107,113,118,122,127,137,139,149,151,197,

%T 206,211,214,223,226,227,229,236,244,278,298,331,347,349,353,359,383,

%U 412,421,422,428,439,452,454,461,479,491,509,523,529,541,556,557,563,569

%N Numbers k which cannot be expressed as the sum of two positive integers k = a + b with a < b so that the continued fraction for a/b has all its partial quotients equal to 1 or 2.

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F21, p. 395.

%e For k = 4, k = 1 + 3. The only continued fraction is 1/3 = [0; 3] and 4 is a term.

%e For k = 5 we have k = 1 + 4 and k = 2 + 3. The continued fractions are 1/4 = [0;4] and 2/3 = [0; 1, 2]. So, 5 is not a term.

%e For k = 17, 5/12 = [0; 2, 2, 2]. So, 17 is not a term.

%t Select[Range@570,ContainsNone[Union@*Rest@*ContinuedFraction/@Table[k/(#-k),{k,Floor[#/2]}],{{1},{2},{1,2}}]&]

%K nonn

%O 1,2

%A _Giorgos Kalogeropoulos_, Nov 29 2021