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 1-10 of 21 results. Next

A007320 Number of steps needed for juggler sequence (A094683) started at n to reach 1.

Original entry on oeis.org

0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, 9, 3, 9, 3, 11, 6, 6, 6, 9, 6, 6, 6, 8, 6, 8, 3, 17, 3, 14, 3, 5, 3, 6, 3, 6, 3, 6, 3, 11, 5, 11, 5, 11, 5, 11, 5, 5, 5, 11, 5, 11, 5, 5, 3, 5, 3, 11, 3, 14, 3, 5, 3, 8, 3, 8, 3, 19, 3, 8, 3, 10, 8, 8, 8, 11, 8, 10, 8, 11, 8, 11, 8, 11, 8, 8, 8, 11
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

Comments

It is not known if every starting value eventually reaches 1.

Examples

			The trajectory of 1 is 3, 5, 11, 36, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... so a(3) = 6.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 232.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007321, A094683, A094698, A094679, A093685, A094716.

Programs

  • Maple
    A007320 := proc(n)
    local a,ntrack;
    a := 0 ;
    ntrack := n ;
    while ntrack > 1 do
        ntrack := A094683(ntrack) ;
        a := a+1 ;
    end do:
    return a;
end proc: # R. J. Mathar, Apr 19 2013
  • Mathematica
    js[n_] := If[ EvenQ[n], Floor[ Sqrt[n]], Floor[ Sqrt[n^3]]]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; Table[ f[n], {n, 99}] (* Robert G. Wilson v, Jun 10 2004 *)

Extensions

Corrected and extended by Jason Earls, Jun 09 2004

A094716 Largest value in trajectory of n under the juggler map of A094683.

Original entry on oeis.org

0, 1, 2, 36, 4, 36, 6, 18, 8, 140, 36, 36, 36, 46, 36, 58, 16, 70, 18, 140, 20, 140, 22, 110, 24, 52214, 36, 140, 36, 156, 36, 172, 36, 2598, 36, 2978, 36, 24906114455136, 38, 233046, 40, 262, 42, 4710, 44, 5222, 46, 322, 48, 6352, 50, 364, 52, 7554, 54, 8210, 56, 430, 58, 946636
Offset: 0

Views

Author

N. J. A. Sloane, Jun 10 2004

Keywords

Comments

Harry J. Smith found that the highest value in the trajectory of 30817 is a number with 45391 digits and the highest value in the trajectory of 48443 is a number with 972463 digits. - Jason Earls, Jun 10 2004

Links

Programs

  • Mathematica
    smx={};Do[a=n;mx=n;While[a!=1,a=If[Mod[a,2]==0,Floor[Sqrt[a]],Floor[a^(3/2)]];If[a>mx,mx=a]];AppendTo[smx,mx],{n,59}];Join[{0},smx] (* James C. McMahon, Apr 15 2025 *)

Extensions

More terms from Jason Earls, Jun 10 2004
a(37) corrected by Labos Elemer, Jun 19 2004
a(49) corrected by Hans Havermann, Dec 07 2017

A094684 Records in A094683.

Original entry on oeis.org

0, 1, 5, 11, 18, 27, 36, 46, 58, 70, 82, 96, 110, 125, 140, 156, 172, 189, 207, 225, 243, 262, 281, 301, 322, 343, 364, 385, 407, 430, 453, 476, 500, 524, 548, 573, 598, 623, 649, 675, 702, 729, 756, 783, 811, 839, 868, 896, 925, 955, 985
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2004

Keywords

Comments

Each odd n in A094683 sets a new record, so this is just a bisection of A094683.

Crossrefs

Cf. A094683.

A094819 Number of steps needed for juggler sequence (A094683) started at 10^n to reach 1.

Original entry on oeis.org

0, 7, 8, 7, 9, 6, 8, 7, 10, 7, 7, 20, 9, 25, 8, 11, 11, 13, 8, 10, 8, 16, 21, 24, 10, 18, 26, 15, 9, 45, 12, 31, 12, 20, 14, 22, 9, 20, 11, 25, 9, 33, 17, 33, 22, 30, 25, 11, 11, 30, 19, 13, 27, 21, 16, 10, 10, 16, 46, 70, 13, 13, 32, 21, 13, 21, 21, 48, 15, 29, 23, 29, 10, 18, 21
Offset: 0

Views

Author

Jason Earls, Jun 12 2004

Keywords

Crossrefs

Cf. A007320, A011557, A094683.

Programs

  • Mathematica
    fj[n_]:=If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]];a094819[n_]:=Length[ NestWhileList[fj,10^n, # != 1 &]] - 1;Array[a094819,75,0] (* James C. McMahon, Apr 18 2025 *)

Formula

a(n) = A007320(A011557(n)). - Michel Marcus, Apr 19 2025

A218335 Even numbers n such that the largest value in trajectory of n under the juggler map of A094683 is greater than n.

Original entry on oeis.org

10, 12, 14, 26, 28, 30, 32, 34, 82, 84, 86, 88, 90, 92, 94, 96, 98, 626, 628, 630, 632, 634, 636, 638, 640, 642, 644, 646, 648, 650, 652, 654, 656, 658, 660, 662, 664, 666, 668, 670, 672, 674, 1090, 1092, 1094, 1096, 1098, 1100, 1102, 1104, 1106, 1108, 1110
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 26 2012

Keywords

Links

Crossrefs

Cf. A094683, A094716.

Programs

  • Mathematica
    lst = {}; Do[n = a; While[True, If[EvenQ[a], a = Floor[a^(1/2)]; If[a == 1, Break[]], a = Floor[a^(3/2)]; If[a > n, AppendTo[lst, n]; Break[]]]], {a, 2, 1110, 2}]; lst

Formula

A number n is in the sequence if and only if n is even and A094716(n) > n.

A094804 Number of primes in the trajectory of n under the juggler map of A094683.

Original entry on oeis.org

0, 0, 1, 4, 1, 3, 1, 2, 1, 2, 4, 2, 4, 2, 4, 2, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 3, 2, 3, 5, 3, 3, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 4, 2, 3, 2, 3, 2, 2, 2, 1, 2, 4, 2, 3, 2, 1, 1, 1, 1, 6, 1, 4, 1, 2, 1, 3, 1, 1, 1, 3, 1, 4, 1, 1, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 5, 2, 2, 2, 4, 2, 1, 4, 3, 4, 2, 4
Offset: 0

Views

Author

Jason Earls, Jun 11 2004

Keywords

Examples

			3 -> 5 -> 11 -> 36 -> 6 -> 2 -> 1, so in this trajectory 3, 5, 11 and 2 are primes, hence a(3) = 4.

Programs

A007321 Number of steps needed for modified juggler sequence (A094685) started at n to reach 1.

Original entry on oeis.org

0, 1, 6, 2, 5, 2, 13, 7, 10, 7, 4, 7, 6, 3, 9, 3, 9, 3, 12, 3, 9, 6, 9, 6, 19, 6, 9, 6, 9, 6, 16, 3, 5, 3, 8, 3, 16, 3, 5, 3, 14, 3, 11, 14, 11, 14, 5, 14, 14, 14, 14, 14, 5, 14, 5, 14, 11, 8, 11, 8, 8, 8, 8, 8, 11, 8, 11, 8, 8, 8, 8, 8, 21, 11, 21, 11, 8, 11, 8, 11, 19, 11, 11, 11, 8, 11, 11, 11, 11
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

Comments

It is not known if every starting value eventually reaches 1.

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A094685, A007320, A094683.

Programs

  • Maple
    f:=proc(n) if n mod 2 = 0 then RETURN(round(sqrt(n))) else RETURN(round(n^(3/2))); fi; end; # corrected by R. J. Mathar, Jul 28 2007
  • Mathematica
    mjs[n_] := If[EvenQ[n], Round[Sqrt[n]], Round[Sqrt[n^3]]]; f[n_] := Length[NestWhileList[mjs, n, # != 1 &]] - 1; Table[ f[n], {n, 90}]

Extensions

More terms from N. J. A. Sloane, Jun 09 2004

A094685 Modified juggler sequence: if n mod 2 = 0 then round(sqrt(n)) else round(n^(3/2)).

Original entry on oeis.org

0, 1, 1, 5, 2, 11, 2, 19, 3, 27, 3, 36, 3, 47, 4, 58, 4, 70, 4, 83, 4, 96, 5, 110, 5, 125, 5, 140, 5, 156, 5, 173, 6, 190, 6, 207, 6, 225, 6, 244, 6, 263, 6, 282, 7, 302, 7, 322, 7, 343, 7, 364, 7, 386, 7, 408, 7, 430, 8, 453, 8, 476, 8, 500, 8, 524, 8, 548, 8, 573, 8, 598, 8, 624, 9, 650
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2004

Keywords

References

  • C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 233.

Links

Crossrefs

Cf. A007320, A007321, A094684, A094683, A094693, A094725.

Programs

  • Maple
    f:=proc(n) if n mod 2 = 0 then RETURN(round(sqrt(n))) else RETURN(round(n^(3/2))); fi; end;
  • Mathematica
    A094685[n_]:=If[Mod[n,2]==0,Round[Sqrt[n]],Round[n^(3/2)]];Array[A094685,76,0] (* James C. McMahon, Apr 15 2025 *)
  • Python
    from gmpy2 import isqrt_rem
def A094685(n):
    i, j = isqrt_rem(n**3 if n % 2 else n)
    return int(i+ int(4*(j-i) >= 1)) # Chai Wah Wu, Aug 17 2016

A095396 Modified juggler map: for even numbers: a(n) = floor(n^(2/3)) and for odd n: a(n) = floor(n^(3/2)) = floor(sqrt(n^3)).

Original entry on oeis.org

1, 1, 5, 2, 11, 3, 18, 4, 27, 4, 36, 5, 46, 5, 58, 6, 70, 6, 82, 7, 96, 7, 110, 8, 125, 8, 140, 9, 156, 9, 172, 10, 189, 10, 207, 10, 225, 11, 243, 11, 262, 12, 281, 12, 301, 12, 322, 13, 343, 13, 364, 13, 385, 14, 407, 14, 430, 14, 453, 15, 476, 15, 500, 16, 524, 16, 548, 16
Offset: 1

Views

Author

Labos Elemer, Jun 18 2004

Keywords

Comments

Parallel to A094683: values for odd arguments are same, for even not necessarily so.

Links

Crossrefs

Cf. A000196, A007320, A048766, A094683, A094716, A095397, A095398.

Programs

  • Mathematica
    d[x_]:=d[x]=(1-Mod[x, 2])*Floor[N[x^(2/3), 50]] +Mod[x, 2]*Floor[N[x^(3/2), 50]];d[1]=1; Table[d[w], {w, 1, 150}]
Table[If[EvenQ[n],Floor[n^(2/3)],Floor[n^(3/2)]],{n,70}] (* Harvey P. Dale, Dec 28 2018 *)
  • Scheme
    (define (A095396 n) (if (even? n) (A048766 (* n n)) (A000196 (* n n n)))) ;; Antti Karttunen, May 28 2017

Formula

For even n: a(n) = A048766(n^2), for odd n: a(n) = A000196(n^3). - Antti Karttunen, May 28 2017

Extensions

Name simplified by Antti Karttunen, May 28 2017

A380891 If n mod 2 = 0 then a(n) = floor(n^(1/3)) else a(n) = floor(n^(4/3)).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 13, 2, 18, 2, 24, 2, 30, 2, 36, 2, 43, 2, 50, 2, 57, 2, 65, 2, 73, 2, 81, 3, 89, 3, 97, 3, 105, 3, 114, 3, 123, 3, 132, 3, 141, 3, 150, 3, 160, 3, 169, 3, 179, 3, 189, 3, 199, 3, 209, 3, 219, 3, 229, 3, 240, 3, 250, 4, 261, 4, 272
Offset: 0

Views

Author

Vikram Prasad, Feb 08 2025

Keywords

Crossrefs

Interspersion of A048766 and A129011.
Cf. A094683, A381246, A383135.

Programs

  • Mathematica
    a[n_]:=If[Mod[n,2]==0,Floor[n^(1/3)],Floor[n^(4/3)]]; (* James C. McMahon, Apr 11 2025 *)
  • Python
    import gmpy2
def a(n): return int(gmpy2.iroot(n**4 if n&1 else n, 3)[0])
Showing 1-10 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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