A036412 -id:A036412 - 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 *)

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 *)

A036416 Number of empty intervals when fractional_part(i*Pi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 4, 5, 7, 7, 7, 8, 9, 10, 11, 14, 12, 12, 14, 13, 14, 16, 21, 15, 17, 18, 16, 18, 19, 21, 20, 20, 21, 20, 22, 22, 23, 21, 22, 22, 23, 24, 25, 27, 21, 24, 24, 23, 25, 25, 28, 22, 25, 24, 26, 25, 25, 27, 21, 25, 24, 26, 24, 25, 25, 21, 24, 22, 23, 22

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.

Cf. A036417 (positions of 0).

Cf. similar sequences with other constants: A036412 (e), A036414 (phi), A046157 (gamma).

Table[Length@Complement[Range[n] - 1, Floor[n*FractionalPart[Pi*Range[n]]]], {n, 75}] (* Ivan Neretin, Jan 23 2018 *)

A036413 Values of k for which there are no empty intervals when fractional_part(m*e) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 32, 35, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959

Offset: 1

Eric W. Weisstein

The sequence is infinite since it contains all denominators of continued fraction convergents of e. The next such denominator is 190435, but it has yet to be proved that there are no other terms before it. - Ivan Neretin, Jan 23 2018
a(17) = 190435 verified as the next term. - Sean A. Irvine, Oct 31 2020
No other terms < 10^6. - Eric W. Weisstein, Apr 21 2024

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

Cf. A036412.

Cf. A007677 (denominators of convergents to e).

With[{f = FractionalPart[E 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(17) from Sean A. Irvine, Oct 31 2020

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

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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.

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

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A036416 Number of empty intervals when fractional_part(i*Pi) for i = 1, ..., n is plotted along [ 0, 1 ] subdivided into n equal regions.

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A036413 Values of k for which there are no empty intervals when fractional_part(m*e) for m = 1, ..., k is plotted along [ 0, 1 ] subdivided into k equal regions.

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