cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Eric W. Weisstein

Keywords

References

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

Links

Crossrefs

Cf. A036415 (positions of 0).
Cf. similar sequences with other constants: A036412 (e), A036416 (Pi), A046157 (gamma).

Programs

  • Mathematica
    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

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A036417 (positions of 0).
Cf. similar sequences with other constants: A036412 (e), A036414 (phi), A046157 (gamma).

Programs

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

A046158 Values of n for which there are no empty intervals when frac(m*gamma) for m = 1, ..., n is plotted along [0, 1] subdivided into n equal regions.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 12, 19, 26, 97, 123, 149, 272, 395, 5258, 5653, 26685, 31943, 58628, 500967, 559595
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

No others with n <= 10^6. - Eric W. Weisstein, Apr 26 2024

Links

Crossrefs

Cf. A046157.
Cf. A046115 (denominators of the convergents to the Euler-Mascheroni constant).
Corresponding sequences for other constants: A036413 (e), A036415 (phi), A036417 (Pi).

Programs

  • Mathematica
    With[{f = FractionalPart[EulerGamma 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(19) from Sean A. Irvine, Nov 01 2020
a(20)-a(21) from Eric W. Weisstein, Apr 26 2024

A082385 For each n append T(n), T(T(n)), T^3(n), ..., T^r(n), where T(n) = A055012(n) and r is the smallest integer such that T^r(n) is one of the following numbers: 1, 55, 136, 153, 160, 370, 371, 407, 919.

Original entry on oeis.org

1, 8, 512, 134, 92, 737, 713, 371, 27, 351, 153, 64, 280, 520, 133, 55, 125, 134, 92, 737, 713, 371, 216, 225, 141, 66, 432, 99, 1458, 702, 351, 153, 343, 118, 514, 190, 730, 370, 512, 134, 92, 737, 713, 371, 729, 1080, 513, 153, 1, 2, 8, 512, 134
Offset: 1

Views

Author

Cino Hilliard, Apr 13 2003

Keywords

Comments

Conjecture: The sequence always terminates with one of the following:(tested to n=1000000) 1,55,136,153,160,370,371,407,919 which eventually loop back to themselves. 1,153,370,371,407 loop back in 1 step and are the sum of the cubes of their digits. The others are 55,250,133,55. 136,244,136. 160,217,352,160. 919,1459,919. A046156, A046157 indicate this as a limit of possibilities of numbers that cubed digital roots roll back to the origional number. Proof? - Cino Hilliard, Apr 13 2003 Proof: In A055012 T. D. Noe notes that for n > 1999, A055012(n) < n. This means that by repeatedly applying A055012, we eventually reach a number smaller than 2000. As checked by Cino Hilliard, all numbers below 10^6 end in one of the listed cycles. - Stefan Steinerberger, Sep 05 2007

Crossrefs

Cf. A046156, A055012.

Programs

  • Mathematica
    a = {}; For[n = 1, n < 9, n++, j = Plus @@ IntegerDigits[n]^3; AppendTo[a, j]; While[ !MemberQ[{1, 55, 136, 153, 160, 370, 371, 407, 919}, j], j = Plus @@ (IntegerDigits[j]^3); AppendTo[a, j]]]; a
  • PARI
    digitcube2(m) = {y=0; for(x=1,m, digitcube(x) ) } digitcube(n) = { while(1, s=0; while(n > 0, d=n%10; s = s+d*d*d; n=floor(n/10); ); print1(s" "); if(s==1 || s==55 || s==153 || s==160 || s==370 || s==371 || s==407 || s==919 || s==136,break); n=s;) }

Extensions

Edited by Stefan Steinerberger, Sep 05 2007
Showing 1-5 of 5 results.

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