A243846 -id:A243846 - cp's OEIS frontend

A335505 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 5*x(j-1) by deleting all occurrences of the digit k in base n.

0, 1, 1, 6, 1, 1, 4, 1, 1, 2, 1, 1, 0, 0, 0, 6, 60, 6, 30, 2, 1, 48, 2, 3, 12, 0, 1, 6, 156, 14, 22, 2, 18, 1, 34, 78, 12, 36, 3, 48, 0, 1, 138, 198, 10, 684, 1, 1, 2, 20, 1, 2, 0, 22, 1872, 495, 2, 50, 315, 0, 1, 405, 245, 2780, 0, 1440

Offset: 1

Pontus von Brömssen, Jun 13 2020

T(1,0) = 0 is defined in order to make the triangle of numbers regular.

T(n,k) = 1 whenever k is a power of 5 and k > 1.

			Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9
  -----------------------------------------------------
   1:    0
   2:    1    1
   3:    6    1    1
   4:    4    1    1    2
   5:    1    1    0    0    0
   6:    6   60    6   30    2    1
   7:   48    2    3   12    0    1    6
   8:  156   14   22    2   18    1   34   78
   9:   12   36    3   48    0    1  138  198   10
  10:  684    1    1    2   20    1    2    0   22 1872
T(10,7) = 0 because A335506 never enters a cycle.

Pontus von Brömssen, Rows n = 1..32, flattened

Cf. A335502, A335503, A335504, A335506.

Cf. A243846, A306569, A306773.

from sympy.ntheory.factor_ import digits
from functools import reduce
def drop(x,n,k):
  # Drop all digits k from x in base n.
  return reduce(lambda x,j:n*x+j if j!=k else x,digits(x, n)[1:],0)
def cycle_length(n,k,m):
  # Brent's algorithm for finding cycle length.
  # Note: The function may hang if the sequence never enters a cycle.
  if (m,n,k)==(5,10,7):
    return 0 # A little cheating; see A335506.
  p=1
  length=0
  tortoise=hare=1
  nz=0
  while True:
    hare=drop(m*hare,n,k)
    while hare and hare%n==0:
      hare//=n
      nz+=1 # Keep track of the number of trailing zeros.
    length+=1
    if tortoise==hare:
      break
    if p==length:
      tortoise=hare
      nz=0
      p*=2
      length=0
  return length if not nz else 0
def A335505(n,k):
  return cycle_length(n,k,5) if n>1 else 0

A335502 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 2*x(j-1) by deleting all occurrences of the digit k in base n.

Original entry on oeis.org

0, 1, 1, 4, 1, 1, 2, 1, 1, 0, 4, 1, 1, 3, 1, 12, 2, 1, 6, 1, 4, 78, 1, 1, 6, 1, 3, 6, 3, 1, 1, 0, 1, 0, 0, 0, 6, 1, 1, 18, 1, 4, 36, 4, 1, 36, 4, 1, 4, 1, 8, 4, 72, 1, 540, 100, 1, 1, 16, 1, 4, 17, 0, 1, 8, 4, 90, 2, 1, 12, 1, 4, 14, 6, 1, 4, 4, 240

Offset: 1

Pontus von Brömssen, Jun 13 2020

T(1,0) = 0 is defined in order to make the triangle of numbers regular.
One way of getting T(n,k) = 0 is to have x(j) = x(i)*n^e for some j > i and e > 0. For k < n <= 48, this is the only way to get T(n,k) = 0 (but see A335506 for another situation where the x-sequence is not periodic).
T(n,k) = 1 whenever k is a power of 2 and k > 1.
It seems that k = 0 and k = n-1 often lead to particularly long cycles.

			Triangle begins:
   n\k  0   1   2   3   4   5   6   7   8   9  10  11
  ---------------------------------------------------
   1:   0
   2:   1   1
   3:   4   1   1
   4:   2   1   1   0
   5:   4   1   1   3   1
   6:  12   2   1   6   1   4
   7:  78   1   1   6   1   3   6
   8:   3   1   1   0   1   0   0   0
   9:   6   1   1  18   1   4  36   4   1
  10:  36   4   1   4   1   8   4  72   1 540
  11: 100   1   1  16   1   4  17   0   1   8   4
  12:  90   2   1  12   1   4  14   6   1   4   4 240
For n = 10 and k = 5, the x-sequence starts 1, 2, 4, 8, 16, 32, 64, 128, 26, 2, and then repeats with a period of 8, so T(10,5) = 8.
T(10,0) = 36, because A242350 eventually enters a cycle of length 36.
For n=11 and k=7, the x-sequence starts (in base 11) 1, 2, 4, 8, 15, 2A, 59, 10. Disregarding trailing zeros, the sequence then repeats with period 7 and x(i+7*j) = x(i)*11^j for positive i and j. The x-sequence itself is therefore not eventually periodic, so T(11,7)=0.

Pontus von Brömssen, Rows n = 1..48, flattened

Cf. A242350.
Cf. A335503, A335504, A335505, A335506.
Cf. A243846, A306569, A306773.

from sympy.ntheory.factor_ import digits
from functools import reduce
def drop(x,n,k):
  # Drop all digits k from x in base n.
  return reduce(lambda x,j:n*x+j if j!=k else x,digits(x, n)[1:],0)
def cycle_length(n,k,m):
  # Brent's algorithm for finding cycle length.
  # Note: The function may hang if the sequence never enters a cycle.
  if (m,n,k)==(5,10,7):
    return 0 # A little cheating; see A335506.
  p=1
  length=0
  tortoise=hare=1
  nz=0
  while True:
    hare=drop(m*hare,n,k)
    while hare and hare%n==0:
      hare//=n
      nz+=1 # Keep track of the number of trailing zeros.
    length+=1
    if tortoise==hare:
      break
    if p==length:
      tortoise=hare
      nz=0
      p*=2
      length=0
  return length if not nz else 0
def A335502(n,k):
  return cycle_length(n,k,2) if n>1 else 0

A335503 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 3*x(j-1) by deleting all occurrences of the digit k in base n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 12, 1, 2, 1, 28, 1, 0, 1, 2, 64, 1, 2, 1, 2, 4, 60, 4, 2, 1, 4, 0, 2, 54, 1, 2, 1, 62, 16, 2, 48, 2, 1, 0, 1, 0, 0, 0, 0, 0, 80, 40, 4, 1, 20, 2000, 60, 72, 4, 1, 40, 20, 5, 1, 85, 240, 5, 5, 20, 1, 320, 1260, 128, 2, 1, 272, 4, 2, 48, 68, 1, 20, 1440

Offset: 1

Pontus von Brömssen, Jun 13 2020

T(1,0) = 0 is defined in order to make the triangle of numbers regular.

T(n,k) = 1 whenever k is a power of 3 and k>1.

			Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9   10   11
  ---------------------------------------------------------------
   1:    0
   2:    1    1
   3:    1    1    0
   4:   12    1    2    1
   5:   28    1    0    1    2
   6:   64    1    2    1    2    4
   7:   60    4    2    1    4    0    2
   8:   54    1    2    1   62   16    2   48
   9:    2    1    0    1    0    0    0    0    0
  10:   80   40    4    1   20 2000   60   72    4    1
  11:   40   20    5    1   85  240    5    5   20    1  320
  12: 1260  128    2    1  272    4    2   48   68    1   20 1440
T(10,0) = 80, because A243845 eventually enters a cycle of length 80.

Pontus von Brömssen, Rows n = 1..32, flattened

Cf. A243845.
Cf. A335502, A335504, A335505, A335506.
Cf. A243846, A306569, A306773.

from sympy.ntheory.factor_ import digits
from functools import reduce
def drop(x,n,k):
  # Drop all digits k from x in base n.
  return reduce(lambda x,j:n*x+j if j!=k else x,digits(x, n)[1:],0)
def cycle_length(n,k,m):
  # Brent's algorithm for finding cycle length.
  # Note: The function may hang if the sequence never enters a cycle.
  if (m,n,k)==(5,10,7):
    return 0 # A little cheating; see A335506.
  p=1
  length=0
  tortoise=hare=1
  nz=0
  while True:
    hare=drop(m*hare,n,k)
    while hare and hare%n==0:
      hare//=n
      nz+=1 # Keep track of the number of trailing zeros.
    length+=1
    if tortoise==hare:
      break
    if p==length:
      tortoise=hare
      nz=0
      p*=2
      length=0
  return length if not nz else 0
def A335503(n,k):
  return cycle_length(n,k,3) if n>1 else 0

A335504 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 4*x(j-1) by deleting all occurrences of the digit k in base n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 2, 24, 2, 2, 1, 16, 18, 1, 6, 1, 42, 33, 1, 1, 15, 1, 24, 3, 3, 1, 1, 0, 1, 0, 0, 0, 3, 1, 195, 27, 1, 465, 147, 2, 6, 1002, 18, 4, 42, 1, 66, 2, 10, 10, 738, 1660, 25, 5, 180, 1, 2, 15, 35, 150, 4, 1490

Offset: 1

Pontus von Brömssen, Jun 13 2020

T(1,0) = 0 is defined in order to make the triangle of numbers regular.

T(n,k) = 1 whenever k is a power of 4 and k>1.

			Triangle begins:
   n\k   0    1    2    3    4    5    6    7    8    9
  -----------------------------------------------------
   1:    0
   2:    1    1
   3:    2    1    1
   4:    1    1    0    0
   5:    2   24    2    2    1
   6:   16   18    1    6    1   42
   7:   33    1    1   15    1   24    3
   8:    3    1    1    0    1    0    0    0
   9:    3    1  195   27    1  465  147    2    6
  10: 1002   18    4   42    1   66    2   10   10  738

Pontus von Brömssen, Rows n = 1..32, flattened

Cf. A335502, A335503, A335505, A335506.

Cf. A243846, A306569, A306773.

from sympy.ntheory.factor_ import digits
from functools import reduce
def drop(x,n,k):
  # Drop all digits k from x in base n.
  return reduce(lambda x,j:n*x+j if j!=k else x,digits(x, n)[1:],0)
def cycle_length(n,k,m):
  # Brent's algorithm for finding cycle length.
  # Note: The function may hang if the sequence never enters a cycle.
  if (m,n,k)==(5,10,7):
    return 0 # A little cheating; see A335506.
  p=1
  length=0
  tortoise=hare=1
  nz=0
  while True:
    hare=drop(m*hare,n,k)
    while hare and hare%n==0:
      hare//=n
      nz+=1 # Keep track of the number of trailing zeros.
    length+=1
    if tortoise==hare:
      break
    if p==length:
      tortoise=hare
      nz=0
      p*=2
      length=0
  return length if not nz else 0
def A335504(n,k):
  return cycle_length(n,k,4) if n>1 else 0

A243845 Numbers generated by recursive procedure a(n) = nozero(a(n-1) * 3), in which the function nozero(x) removes all zeros from x, starting with a(1) = 1.

Original entry on oeis.org

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 5949, 17847, 53541, 16623, 49869, 14967, 4491, 13473, 4419, 13257, 39771, 119313, 357939, 173817, 521451, 1564353, 469359, 14877, 44631, 133893, 41679, 12537, 37611, 112833, 338499, 115497, 346491, 139473, 418419

Offset: 1

Anthony Sand, Jun 12 2014

Numbers in the following sequence: Let a(1) = 1, then begin the recursive sequence a(n) = nozero(a(n-1) * 3), where the function nozero(x) removes all zeros from x.
The sequence returns standard powers of 3 until step 11, where a(11) = nozero(19683 * 3) = nozero(59049) = 5949.
At step 28, a(28) = nozero(469359 * 3) = nozero(1408077) = 14877. At step 108, a(108) = nozero(4959 * 3) = 14877. Therefore a(28) = a(108) and the sequence repeats. Because this is the first instance where a member of this sequence is repeated one has a(n + L) = a(n) for n >= 28 with the primitive (least) period length L = 108 - 28 = 80.

			a(2) = nozero(3*a(1)) = nozero(3) = 3.

Anthony Sand, Table of n, a(n) for n = 1..108
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000244, A004719, A242350, A243846.

NestList[FromDigits@ DeleteCases[IntegerDigits[3 #], ?(# == 0 &)] &, 1, 38] (* _Michael De Vlieger, Jun 27 2020 *)

L=[1]
for i in [1..108]:
    T=(3*L[i-1]).digits(base=10)
    TT=filter(lambda a: a != 0, T)
    L.append(sum(TTi*10^i for i, TTi in enumerate(TT)))
L # - Tom Edgar, Jun 17 2014

a(n) = A004719(a(n-1) * 3) for n>1, a(1) = 1.

Edited: Name, comments and formula reformulated. - Wolfdieter Lang, Jul 13 2014

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A335505 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 5*x(j-1) by deleting all occurrences of the digit k in base n.

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A335502 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 2*x(j-1) by deleting all occurrences of the digit k in base n.

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A335503 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 3*x(j-1) by deleting all occurrences of the digit k in base n.

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Python

A335504 Triangle read by rows, 0 <= k < n, n >= 1: T(n,k) is the eventual period of the sequence x(j) (or 0 if x(j) never enters a cycle) defined as follows: x(0) = 1 and for j > 1 x(j) is obtained from 4*x(j-1) by deleting all occurrences of the digit k in base n.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A243845 Numbers generated by recursive procedure a(n) = nozero(a(n-1) * 3), in which the function nozero(x) removes all zeros from x, starting with a(1) = 1.

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