A094684 -id:A094684 - cp's OEIS frontend

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

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

N. J. A. Sloane, Jun 09 2004

nonn

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

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

Karl V. Boddy, Table of n, a(n) for n = 0..10000
Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.
Harry J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence
Wikipedia, Juggler sequence

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

Cf. A000093, A000196.

A094683 :=proc(n) if n mod 2 = 0 then RETURN(floor(sqrt(n))) else RETURN(floor(n^(3/2))); end if; end proc;

Table[If[EvenQ[n], Floor[Sqrt[n]], Floor[n^(3/2)]], {n, 0, 100}] (* Indranil Ghosh, Apr 07 2017 *)

a(n) = if(n%2,sqrtint(n^3), sqrtint(n)); \\ Indranil Ghosh, Apr 08 2017

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

from math import isqrt
def A094683(n): return isqrt(n**3 if n % 2 else n) # Chai Wah Wu, Feb 18 2022

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

N. J. A. Sloane, Jun 09 2004

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Vikram Prasad and M. A. Prasad, Estimates of the maximum excursion constant and stopping constant of juggler-like sequences, ResearchGate, 2025.
Harry J. Smith, Juggler Sequence

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

f:=proc(n) if n mod 2 = 0 then RETURN(round(sqrt(n))) else RETURN(round(n^(3/2))); fi; end;

A094685[n_]:=If[Mod[n,2]==0,Round[Sqrt[n]],Round[n^(3/2)]];Array[A094685,76,0] (* James C. McMahon, Apr 15 2025 *)

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

A143745 The next largest juggler number.

Original entry on oeis.org

1, 2, 36, 140, 52214, 24906114455136, 202924588924125339424550328

Offset: 1

Harry J. Smith, Oct 08 2008

nonn

The juggler sequence: begin with a starting value x and if x is even, x <- floor(sqrt(x)) and if x is odd, x <- floor(sqrt(x^3)) and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

I have a b-file for this sequence for n=1,...,19 (all known values), but some a(n) values are much larger than 1000 digits.

			24906114455136 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

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

Harry J. Smith, Table of n, a(n) for n = 1..9
H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

A143742 Starting values that produce a larger juggler number than smaller starting values.

Original entry on oeis.org

1, 2, 3, 9, 25, 37, 113, 173, 193, 2183, 11229, 15065, 15845, 30817, 48443, 275485, 1267909, 2264915, 5812827, 7110201

Offset: 1

Harry J. Smith, Oct 06 2008

nonn

The juggler sequence: begin with a starting value x and if x is even, x -> [sqrt(x)] and if x is odd, x -> [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

			37 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

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

Eric Weisstein's World of Mathematics, Juggler Sequence
H. J. Smith, Juggler Sequence

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

DeleteDuplicates[Table[{n,Max[NestWhileList[If[EvenQ[#],Floor[Sqrt[#]],Floor[Sqrt[#^3]]]&,n,#!=1&]]},{n,50000}],GreaterEqual [#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 15 terms of the sequence. *) (* Harvey P. Dale, Dec 21 2024 *)

Comment clarified by Harvey P. Dale, Dec 21 2024

A143743 The number of digits in the next largest juggler number.

Original entry on oeis.org

1, 1, 2, 3, 5, 14, 27, 82, 271, 5929, 8201, 11723, 23889, 45391, 972463, 1909410, 1952329, 2855584, 7996276

Offset: 1

Harry J. Smith, Oct 08 2008

The juggler sequence: begin with a starting value x and if x is even, x <- [sqrt(x)] and if x is odd, x <- [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

			14 is in the sequence because starting at 37 the juggler sequence maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

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

H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

A143744 The number of steps needed to generate the next largest juggler number.

Original entry on oeis.org

0, 0, 3, 2, 3, 8, 9, 17, 47, 32, 54, 25, 43, 39, 60, 148, 99, 89, 67

Offset: 1

Harry J. Smith, Oct 08 2008

nonn

The juggler sequence: begin with a starting value x and if x is even, x <- [sqrt(x)] and if x is odd, x <- [sqrt(x^3)] and repeat until x = 1, save the starting value, max x and the number of steps needed to reach it.

			8 is in the sequence because starting at 37 the juggler sequences maxes out at 24906114455136, a 14-digit number, after 8 steps. This is the largest juggler number found for starting values less than or equal to 37.

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

Eric Weisstein's World of Mathematics, Juggler Sequence
H. J. Smith, Juggler Sequence

Cf. A007320, A094670, A094679, A094683, A094684, A095908.

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

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

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A143745 The next largest juggler number.

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A143742 Starting values that produce a larger juggler number than smaller starting values.

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A143743 The number of digits in the next largest juggler number.

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Author

Keywords

Comments

Examples

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Crossrefs

A143744 The number of steps needed to generate the next largest juggler number.

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Author

Keywords

Comments

Examples

References

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Crossrefs