A036415 Values of k for which there are no empty intervals when fractional part(m*phi) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• more

Comments

The sequence contains the Fibonacci numbers A000045.

If fact, for all known terms with n > 10, a(n) = A000045(n-3). - Eric W. Weisstein, Apr 17 2024

No others terms with n <= 10^6. - Eric W. Weisstein, Apr 28 2024

References

• H. Steinhaus, Mathematical Snapshots, 3rd American ed., New York: Oxford University Press, pp. 48-49, 1983.

Links

• Eric Weisstein's World of Mathematics, Equidistributed Sequence.
• Eric Weisstein's World of Mathematics, Golden Ratio.

Crossrefs

Cf. A000045, A036414.

Programs

Mathematica

With[{f = FractionalPart[GoldenRatio Range[1000]]}, Position[Table[Count[BinCounts[Take[f, n], {0., 1, 1/n}], 0], {n, Length[f]}], 0]] // Flatten (* Eric W. Weisstein, Apr 27 2024 *)

Extensions

a(27)-a(29) from Sean A. Irvine, Oct 31 2020

a(30)-a(31) from Eric W. Weisstein, Apr 18-19 2024

a(32)-a(33) from Eric W. Weisstein, Apr 28 2024