A349765 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.

1, 2, 4, 23, 37, 53, 59, 61, 83, 103, 107, 113, 118, 122, 127, 137, 139, 149, 151, 197, 206, 211, 214, 223, 226, 227, 229, 236, 244, 278, 298, 331, 347, 349, 353, 359, 383, 412, 421, 422, 428, 439, 452, 454, 461, 479, 491, 509, 523, 529, 541, 556, 557, 563, 569

Offset

1,2

References

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

Links

Table of n, a(n) for n=1..55.

Example

For k = 4, k = 1 + 3. The only continued fraction is 1/3 = [0; 3] and 4 is a term.
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.
For k = 17, 5/12 = [0; 2, 2, 2]. So, 17 is not a term.

Mathematica

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

Crossrefs

Sequence in context: A192332 A324603 A322520 * A018279 A307459 A133834

Adjacent sequences: A349762 A349763 A349764 * A349766 A349767 A349768

Keyword

nonn

Author

Giorgos Kalogeropoulos, Nov 29 2021

Status

approved