A036416 -id:A036416 - cp's OEIS frontend

A046157 Number of empty intervals when fractional_part(i*gamma) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions, where gamma is the Euler-Mascheroni constant.

0, 0, 0, 1, 0, 0, 0, 1, 2, 2, 3, 0, 3, 5, 3, 3, 3, 2, 0, 3, 2, 2, 2, 1, 1, 0, 2, 3, 4, 3, 3, 3, 7, 6, 7, 7, 8, 8, 13, 10, 10, 10, 12, 11, 6, 13, 14, 14, 14, 14, 15, 26, 16, 19, 14, 17, 18, 18, 19, 18, 20, 18, 21, 18, 20, 20, 21, 21, 19, 21, 15, 20, 23, 21, 21, 20, 22, 26, 21, 22, 20

Offset: 1

Eric W. Weisstein

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Equidistributed Sequence.

Cf. A046158 (positions of 0).

Similar sequences with other constants: A036412 (e), A036414 (phi), A036416 (Pi).

Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[EulerGamma*Range[n]]]], {n, 81}] (* Ivan Neretin, Jan 23 2018 *)

A036412 Number of empty intervals when fractional_part(i*e) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 3, 1, 4, 4, 7, 5, 5, 6, 4, 4, 6, 7, 6, 8, 5, 2, 6, 4, 5, 3, 4, 3, 0, 3, 2, 0, 3, 3, 3, 0, 4, 4, 5, 6, 5, 6, 7, 8, 8, 8, 9, 8, 8, 7, 8, 8, 8, 9, 8, 8, 7, 6, 7, 6, 1, 5, 4, 4, 3, 2, 2, 0, 5, 4, 3, 5, 5, 6, 2, 8, 9, 9, 10, 11, 9, 11, 13, 16, 14, 16, 16, 17, 17, 18, 18, 20

Offset: 1

Eric W. Weisstein

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Equidistributed Sequence.
Eric Weisstein's World of Mathematics, Golden Ratio.

Cf. A036413 (positions of 0).
Cf. similar sequences with other constants: A036414 (phi), A036416 (Pi), A046157 (gamma).
Cf. A001113.

Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[E*Range[n]]]], {n, 95}] (* Ivan Neretin, Jan 23 2018 *)

A036414 Number of empty intervals when fractional_part(i*phi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, 0, 2, 3, 1, 2, 0, 3, 2, 4, 3, 1, 3, 3, 4, 3, 2, 4, 5, 0, 4, 5, 4, 8, 6, 6, 5, 2, 5, 5, 5, 5, 8, 5, 5, 4, 8, 6, 6, 5, 0, 6, 7, 8, 7, 6, 8, 8, 11, 9, 8, 10, 9, 4, 9, 9, 9, 8, 8, 9, 8, 12, 8, 8, 10, 9, 6, 9, 8, 11, 10, 8, 10, 10, 0, 10, 11, 9, 12, 12, 14

Offset: 1

Eric W. Weisstein

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

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Equidistributed Sequence
Eric Weisstein's World of Mathematics, Golden Ratio

Cf. A036415 (positions of 0).

Cf. similar sequences with other constants: A036412 (e), A036416 (Pi), A046157 (gamma).

Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[GoldenRatio*Range[n]]]], {n, 95}] (* Ivan Neretin, Jan 23 2018 *)
Table[Count[BinCounts[FractionalPart[GoldenRatio Range[n]], {0, 1, 1/n}], 0], {n, 95}] (* Eric W. Weisstein, Apr 17 2024 *)

A036417 Values of k for which there are no empty intervals when fractional part(m*Pi) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

Original entry on oeis.org

1, 6, 7, 106, 112, 113, 33102, 33215, 66317, 99532, 165849, 265381, 364913

Offset: 1

Eric W. Weisstein

Appears to include all the denominators of the convergents of Pi A002486. - Eric W. Weisstein, Apr 18 2024

No other terms with n <= 10^6. - Eric W. Weisstein, Apr 27 2024

Eric Weisstein's World of Mathematics, Equidistributed Sequence.
Eric Weisstein's World of Mathematics, Pi.

Cf. A036416.

Cf. A002486 (denominators of the convergents of Pi).

With[{f = FractionalPart[Pi 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 *)

a(9)-a(10) from Sean A. Irvine, Oct 31 2020

a(11)-a(13) from Eric W. Weisstein, Apr 18-19 2024

cp's OEIS Frontend

A046157 Number of empty intervals when fractional_part(i*gamma) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions, where gamma is the Euler-Mascheroni constant.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A036412 Number of empty intervals when fractional_part(i*e) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A036414 Number of empty intervals when fractional_part(i*phi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

A036417 Values of k for which there are no empty intervals when fractional part(m*Pi) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions