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 1-8 of 8 results.

A094696 Records in A007321.

Original entry on oeis.org

0, 1, 6, 13, 19, 21
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2004

Keywords

Comments

Comment from R. J. Mathar, Jul 28 2007: The records A094696 and A095911 are probably the same sequence.

Crossrefs

Cf. A007321, A094697.

A094697 Where records occur in A007321.

Original entry on oeis.org

1, 2, 3, 7, 25, 73
Offset: 1

Views

Author

N. J. A. Sloane, Jun 09 2004

Keywords

Comments

Comment from R. J. Mathar, Jul 28 2007: The records A094697 and A095910 are probably the same sequence.

Crossrefs

Cf. A007321, A094696.

A094683 Juggler sequence: if n mod 2 = 0 then floor(sqrt(n)) else floor(n^(3/2)).

Original entry on oeis.org

0, 1, 1, 5, 2, 11, 2, 18, 2, 27, 3, 36, 3, 46, 3, 58, 4, 70, 4, 82, 4, 96, 4, 110, 4, 125, 5, 140, 5, 156, 5, 172, 5, 189, 5, 207, 6, 225, 6, 243, 6, 262, 6, 281, 6, 301, 6, 322, 6, 343, 7, 364, 7, 385, 7, 407, 7, 430, 7, 453, 7, 476, 7, 500, 8, 524, 8, 548, 8, 573, 8, 598, 8, 623, 8, 649
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2004

Keywords

Comments

Interspersion of A000093 and A000196. - Michel Marcus, Nov 11 2013

References

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

Links

Crossrefs

Cf. A007320, A007321, A094684, A094685, A094716.
Cf. A000093, A000196.

Programs

  • Maple
    A094683 :=proc(n) if n mod 2 = 0 then RETURN(floor(sqrt(n))) else RETURN(floor(n^(3/2))); end if; end proc;
  • Mathematica
    Table[If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]], {n, 0, 100}] (* Indranil Ghosh, Apr 07 2017 *)
  • PARI
    a(n) = if(n%2,sqrtint(n^3), sqrtint(n)); \\ Indranil Ghosh, Apr 08 2017
  • Python
    import math
from sympy import sqrt
def a(n): return int(math.floor(sqrt(n))) if n%2 == 0 else int(math.floor(n**(3/2)))
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 08 2017
  • Python
    from math import isqrt
def A094683(n): return isqrt(n**3 if n % 2 else n) # Chai Wah Wu, Feb 18 2022

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

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

A095910 Where the modified Juggler sequence reaches a new record.

Original entry on oeis.org

2, 3, 7, 25, 73, 101, 471, 549, 791, 837, 1199, 2267, 29773, 38653, 273317, 2776203, 3683335, 4975111, 8087437
Offset: 1

Views

Author

Robert G. Wilson v, Jun 14 2004

Keywords

Comments

The modified Juggler sequence: begin with x and if x is even, round(sqrt(x)) -> x and if x is odd, round(sqrt(x3)) -> x and repeat until x = 1, count the iterations.
273317 reaches a maximum of 1640703316...(skip the next 35800 digits)...2531417380 during its trip to 1.
Comment from R. J. Mathar, Jul 28 2007: The records A094697 and A095910 are probably the same sequence.

Crossrefs

Cf. A007321, A095909, A095911.

Programs

  • Mathematica
    mjs[n_] := If[ EvenQ[n], Round[ Sqrt[n]], Round[ Sqrt[n^3]]]; f[n_] := Length[ NestWhileList[mjs, n, # != 1 &]] - 1; a = {0}; Do[ b = f[n]; If[b > a[[ -1]], AppendTo[a, b]], {n, 250000}]; a

Extensions

a(16)-a(19) from Chai Wah Wu, Aug 18 2016

A095911 Number of steps where the modified Juggler sequence reaches a new record (A095910).

Original entry on oeis.org

1, 6, 13, 19, 21, 62, 64, 83, 94, 112, 120, 142, 144, 301, 306, 311, 322, 338, 357
Offset: 1

Views

Author

Robert G. Wilson v, Jun 14 2004

Keywords

Comments

The modified Juggler sequence: begin with x and if x is even, round(sqrt(x)) -> x and if x is odd, round(sqrt(x3)) -> x and repeat until x = 1, count the iterations.
The records A094696 and A095911 are probably the same sequence. - R. J. Mathar, Jul 28 2007

Crossrefs

Cf. A007321, A095909, A095910.

Programs

  • Mathematica
    mjs[n_] := If[ EvenQ[n], Round[ Sqrt[n]], Round[ Sqrt[n^3]]];; f[n_] := Length[ NestWhileList[mjs, n, # != 1 &]] - 1; a = {0}; Do[ b = f[n]; If[b >a[[ -1]] AppendTo[a, b]; Print[n]], {n, 250000}]

Extensions

a(16)-a(19) from Chai Wah Wu, Aug 18 2016

A095909 First occurrence of n in the modified Juggler sequence.

Original entry on oeis.org

2, 4, 14, 11, 5, 3, 8, 35, 15, 9, 43, 19, 7, 41, 173, 31, 693, 123, 25, 437, 73, 2239, 303, 215, 1349, 165, 9549, 875, 655, 3439, 229, 121, 1401, 599, 163, 853, 295, 103, 307, 157, 2067, 841, 26179, 883, 775, 8465, 417, 639, 3099, 1487, 44857, 1275, 1307
Offset: 1

Views

Author

Robert G. Wilson v, Jun 14 2004

Keywords

Comments

The modified Juggler sequence: begin with x and if x is even, round(sqrt(x)) -> x and if x is odd, round(sqrt(x3)) -> x and repeat until x = 1, count the iterations.

Crossrefs

Cf. A007321, A095910, A095911.

Programs

  • Mathematica
    mjs[n_] := If[ EvenQ[n], Round[ Sqrt[n]], Round[ Sqrt[n^3]]]; f[n_] := Length[ NestWhileList[mjs, n, # != 1 &]] - 1; a = Table[0, {66}]; Do[ b = f[n]; If[b < 67 && a[[b]] == 0, a[[b]] = n], {n, 44900}]; a
Showing 1-8 of 8 results.

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

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