A093685 -id:A093685 - cp's OEIS frontend

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

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

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

nonn

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

			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.

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).

Giovanni Resta, Table of n, a(n) for n = 1..10000
H. J. Smith, Juggler Sequence
Eric Weisstein's World of Mathematics, Juggler Sequence
Wikipedia, Juggler sequence
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992

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

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

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 *)

Corrected and extended by Jason Earls, Jun 09 2004

A093684 In binary representation: number of occurrences of n in n!.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 10 2004

nonn

a(A093685(n)) = 0, a(A093686(n)) > 0.

			n=12->'1100', 12!=479001600->'11100100011001111110000000000' with three occurrences of '1100': '.1100....1100....1100........', therefore a(12)=3.

Cf. A036603, A007088, A000142, A092601.

f:= proc(n) local L,Lf;
  L:= convert(convert(n,binary),string);
  Lf:= convert(convert(n!,binary),string);
  nops([StringTools:-SearchAll(L,Lf)])
end proc:
map(f, [$1..100]); # Robert Israel, May 20 2016

non[n_]:=Module[{b=IntegerDigits[n,2],f=IntegerDigits[n!,2]}, Length[ Select[ Partition[ f,Length[b],1],#==b&]]]; Array[non,110] (* Harvey P. Dale, Jun 04 2014 *)

A093711 Numbers k such that in their binary representation not all numbers from 1 to k are contained in k!.

Original entry on oeis.org

5, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 32, 36, 37, 38, 40, 41, 44, 45, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 59, 60, 65, 67, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Reinhard Zumkeller, Apr 11 2004

Amiram Eldar, Table of n, a(n) for n = 1..6227 (terms not exceeding 10^4)

Complement of A093710.
A093685 \ {0} is a subsequence.
Cf. A000142, A007088, A036603, A092601.

A092601(a(n)) < a(n).

A093686 In binary representation: numbers occurring at least once in their factorial.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Reinhard Zumkeller, Apr 10 2004

nonn

A093684(a(n)) > 0, complement of A093685.

Almost all numbers are included -- of the first 1000 numbers, only 14 -- i.e., 5, 11, 13, 15, 17, 31, 37, 55, 81, 164, 395, 513, 517, and 619 -- do not appear. In all likelihood, the density of such exceptions gets even smaller as the numbers get larger. - Harvey P. Dale, May 16 2025

			6!=1*2*3*4*5*6=720 -> '1011010000' where '110'=6 is contained:
'..110.....', therefore 6 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A036603, A007088, A000142.

Select[Range[80],SequenceCount[IntegerDigits[#!,2],IntegerDigits[#,2]]>0&] (* Harvey P. Dale, May 16 2025 *)

A093826 In binary representation: least number, k, which occurs n times in its factorial.

Original entry on oeis.org

5, 1, 16, 12, 49, 58, 60, 110, 209, 117, 240, 430, 255, 1423, 921, 980, 511, 1847, 3737, 3692, 3998, 7265, 15267, 15651, 15722, 31457, 32659, 64248, 57927, 64448, 64171, 250068, 129013, 501578, 256159, 510732, 980930, 979883

Offset: 0

Robert G. Wilson v and Reinhard Zumkeller, Apr 16 2004

Overlapping occurrences are counted. - Michael S. Branicky, May 01 2021

a(47) = 262143. - Michael S. Branicky, May 02 2021

			12!_b = 11100100011001111110000000000 and 12_b = 1100 and the later string appears thrice in the former string.

Cf. A093685.

f[n_] := ToString[ FromDigits[ IntegerDigits[n, 2]]]; g[n_] := Length[ StringPosition[ f[n! ], f[n]]]; a = Table[0, {30}]; Do[ b = g[n]; If[a[[b + 1]] == 0, a[[b + 1]] = n], {n, 29000}]; a

from itertools import count, takewhile
def count_overlaps(subs, s):
  c = i = 0
  while i != -1:
    i = s.find(subs, i)
    if i != -1: c += 1; i += 1
  return c
def afind(limit):
  kfact, adict = 1, dict()
  for k in range(1, limit+1):
    kb, kfact = bin(k)[2:], kfact * k
    kfactb = bin(kfact)[2:]
    n = count_overlaps(kb, kfactb)
    if n not in adict: adict[n] = k
  return [adict[n] for n in takewhile(lambda i: i in adict, count(0))]
print(afind(16000))  # Michael S. Branicky, May 01 2021

a(25)-a(37) from Michael S. Branicky, May 03 2021

cp's OEIS Frontend

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

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A093684 In binary representation: number of occurrences of n in n!.

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A093711 Numbers k such that in their binary representation not all numbers from 1 to k are contained in k!.

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A093686 In binary representation: numbers occurring at least once in their factorial.

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A093826 In binary representation: least number, k, which occurs n times in its factorial.

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