A036447 -id:A036447 - cp's OEIS frontend

A321539 3^n with digits rearranged into nonincreasing order.

1, 3, 9, 72, 81, 432, 972, 8721, 6651, 98631, 99540, 777411, 544311, 9543321, 9987642, 98744310, 76443210, 964321110, 988744320, 7666422111, 8876444310, 65433321000, 99865331100, 98877443211, 988654432221, 988876444320, 9888655432221

Offset: 0

N. J. A. Sloane, Nov 19 2018

Paolo Xausa, Table of n, a(n) for n = 0..1000

The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

Cf. A004186.

A321539[n_]:=FromDigits[ReverseSort[IntegerDigits[3^n]]];Array[A321539,40,0] (* Paolo Xausa, Aug 10 2023 *)

def A321539(n): return int(''.join(sorted(str(3**n),reverse=True))) # Chai Wah Wu, Nov 10 2022

a(n) = A004186(A000244(n)). - Michel Marcus, Nov 10 2022

A321540 3^n with digits rearranged into nondecreasing order.

Original entry on oeis.org

1, 3, 9, 27, 18, 234, 279, 1278, 1566, 13689, 4599, 114777, 113445, 1233459, 2467899, 1344789, 1234467, 11123469, 23447889, 1112246667, 134446788, 12333456, 113356899, 11234477889, 122234456889, 23444678889, 1222345568889, 2445567778899

Offset: 0

N. J. A. Sloane, Nov 19 2018

The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

Cf. A004185.

[Seqint(Reverse(Sort(Intseq(3^n)))):n in [0..35]]; // Vincenzo Librandi, Jan 22 2020

Table[FromDigits[Sort[IntegerDigits[3^n]]], {n, 0, 40}] (* Vincenzo Librandi, Jan 22 2020 *)

def A321540(n): return int(''.join(sorted(str(3**n)))) # Chai Wah Wu, Nov 10 2022

a(n) = A004185(A000244(n)). - Michel Marcus, Nov 10 2022

A321541 a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nonincreasing order.

Original entry on oeis.org

1, 3, 9, 72, 621, 8631, 98532, 996552, 9986652, 99996552, 999986652, 9999996552, 99999986652, 999999996552, 9999999986652, 99999999996552, 999999999986652, 9999999999996552, 99999999999986652, 999999999999996552, 9999999999999986652, 99999999999999996552, 999999999999999986652

Offset: 0

N. J. A. Sloane, Nov 19 2018

In contrast to A321542, this sequence increases forever.
Proof: The terms from a(7) onwards can be described as follows:
3 times the number 9 (2k times) 6552 is 2 9 (2k-1 times) 89656 which becomes 9 (2k times) 86652 when sorted;
then 3 times the number 9 (2k times) 86652 is 2 9 (2k times) 59956 which becomes 9 (2k+2 times) 6552 when sorted. QED

Paolo Xausa, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,1,-10).

The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

NestList[FromDigits[ReverseSort[IntegerDigits[3*#]]] &, 1, 25] (* Paolo Xausa, Aug 02 2024 *)

From Chai Wah Wu, Nov 20 2018: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 9.
G.f.: (118800*x^9 + 8910*x^8 + 8811*x^7 + 12321*x^6 + 2439*x^5 - 78*x^4 - 11*x^3 - 22*x^2 - 7*x + 1)/((x - 1)*(x + 1)*(10*x - 1)). (End)

A045539 Multiply by 5 and reverse.

Original entry on oeis.org

1, 5, 52, 62, 13, 56, 82, 14, 7, 53, 562, 182, 19, 59, 592, 692, 643, 5123, 51652, 62852, 62413, 560213, 5601082, 1450082, 140527, 536207, 5301862, 1390562, 182596, 89219, 590644, 223592, 697111, 5555843, 51297772, 68884652, 62324443, 512226113, 5650311652

Offset: 0

Erich Friedman

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 0..100000

Cf. A036447 (*2), A163632 (*3), A132064 (*4), A132078 (*6), A132114 (*7), A132113 (*8), A133361 (*9).

a:= proc(n) option remember; `if`(n=0, 1,
      (s-> parse(cat(s[-i]$i=1..length(s))))(""||(5*a(n-1))))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 09 2015

a[n_] := a[n] = If[n==0, 1, IntegerReverse[5a[n-1]]];
a /@ Range[0, 40] (* Jean-François Alcover, Jan 01 2021 *)

A132064 Numbers multiplied by 4 and written backwards.

Original entry on oeis.org

1, 4, 61, 442, 8671, 48643, 275491, 4691011, 44046781, 421781671, 4866217861, 44417846491, 469583176771, 4807072338781, 42155398282291, 461921395126861, 4447050855867481, 42996432430288771, 480551127927589171, 4866530171154022291, 46198061648602166491

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 30 2007

			a(4) = reverse(4*a(3)) = reverse(4*reverse(4*a(2))) = reverse(4*reverse(4*reverse(4*a(1)))) = reverse(4*reverse(4*4)) = reverse(4*61) = reverse(244) = 442

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A036447 (*2), A163632 (*3), A045539 (*5), A132078 (*6), A132114 (*7), A132113 (*8), A133361 (*9).

a:= proc(n) option remember; `if`(n=1, 1,
      (s-> parse(cat(s[-i]$i=1..length(s))))(""||(4*a(n-1))))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Apr 09 2015

NestList[IntegerReverse[4#]&,1,20] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Dec 09 2017 *)

a(n) = reverse(4*a(n-1)) where a(1) = 1

Conjecture: a(n)^(1/n) tends to 10. - Vaclav Kotesovec, Jan 03 2020

More terms from Alois P. Heinz, Apr 09 2015

A132078 Multiply previous term by 6 and reverse.

Original entry on oeis.org

1, 6, 63, 873, 8325, 5994, 46953, 817182, 2903094, 46581471, 628884972, 2389033773, 83620243341, 640064127105, 362674830483, 8982898406712, 27204409379835, 10972654622361, 66143772953856, 631327736268693, 8512167146697873, 83278108820037015, 90222029256866994

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 30 2007

			a(4) = reverse(6 * a(3)) = reverse(6 * reverse(6 * a(2))) = reverse(6 * reverse( 6 * reverse(6 * a(1)))) = reverse(6 * reverse(6 * reverse(6))) = reverse(6 * 63) = 873.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..10000

Cf. A036447 (*2), A163632 (*3), A132064 (*4), A045539 (*5), A132114 (*7), A132113 (*8), A133361 (*9).

f:=func; a:=[1]; for n in [2..25] do Append(~a,f(a[n-1]));  end for; a; // Marius A. Burtea, Jan 03 2020

Nest[Append[#,IntegerReverse[6*#[[-1]]]]&,{1},22] (* James C. McMahon, Mar 03 2025 *)

a(n) = reverse(6 * a(n-1)) where a(1) = 1.

Name clarified and terms a(16) and beyond from Andrew Howroyd, Jan 02 2020

A132113 Multiply previous term by 8 and reverse.

Original entry on oeis.org

1, 8, 46, 863, 4096, 86723, 487396, 8619983, 46895986, 888761573, 4852900117, 63900232883, 460368102115, 296184492863, 4092495749632, 65079956993723, 487949556936025, 28845546953093, 447426573467032, 6526377852149753

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 31 2007

			a(4) = reverse(8 * a(3))
= reverse(8 * reverse(8 * a(2)))
= reverse(8 * reverse(8 * reverse(8 * a(1))))
= reverse(8 * reverse(8 * reverse(8)))
= reverse(8 * reverse(8 * 8))
= reverse(8 * 46)
= reverse(368)
= 863.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A036447 (*2), A163632 (*3), A132064 (*4), A045539 (*5), A132078 (*6), A132114 (*7), A133361 (*9).

rev:=proc(n) local nn: nn:=convert(n,base,10): add(nn[j]*10^(nops(nn)-j), j = 1..nops(nn)) end proc: a[1]:=1: for n from 2 to 20 do a[n]:=rev(8*a[n-1]) end do: seq(a[n],n=1..20); # Definition corrected by Emeric Deutsch, Nov 07 2007

NestList[IntegerReverse[8#]&,1,20] (* Harvey P. Dale, Dec 22 2018 *)

a(n) = reverse(8 * a(n-1)) where a(1) = 1.

Conjecture: a(n)^(1/n) tends to 10. - Vaclav Kotesovec, Jan 03 2020

Definition corrected by Emeric Deutsch, Nov 07 2007

Edited by Jon E. Schoenfield, May 10 2019

A132114 Multiply previous term by 7 and reverse.

Original entry on oeis.org

1, 7, 94, 856, 2995, 56902, 413893, 1527982, 47859601, 702710533, 1373798194, 8537856169, 38139946795, 565726979662, 4367588800693, 15840612137503, 125269482488011, 770614773688678, 6470285143034935, 54544210069919254

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 31 2007

			a(4) = reverse(7 * a(3)) = reverse(7 * reverse(7 * a(2))) = reverse(7 * reverse(7 * reverse(7 * a(1)))) = reverse(7 * reverse(7 * reverse(7))) = reverse(7*94) = 856.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..10000

Cf. A036447 (*2), A163632 (*3), A132064 (*4), A045539 (*5), A132078 (*6), A132113 (*8), A133361 (*9).

f:=func; a:=[1]; for n in [2..20] do Append(~a,f(a[n-1]));  end for; a; // Marius A. Burtea, Jan 03 2020

seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=fromdigits(Vecrev(digits(a[n-1]*7)))); a} \\ Andrew Howroyd, Jan 02 2020

a(n) = reverse(7 * a(n-1)) with a(1) = 1.

Name edited by Andrew Howroyd, Jan 02 2020

A133361 Multiply by 9 and reverse.

Original entry on oeis.org

1, 9, 18, 261, 9432, 88848, 236997, 3792312, 80803143, 782822727, 3454045407, 36680468013, 711212421033, 7929871190046, 41401704886317, 358679343516273, 7546461904118223, 70046073175181976, 487736675856414036, 4236277072800369834, 60582330255639462183

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 26 2007

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A036447 (*2), A163632 (*3), A132064 (*4), A045539 (*5), A132078 (*6), A132114 (*7), A132113 (*8).

a:= proc(n) option remember; `if`(n=1, 1,
      (s-> parse(cat(s[-i]$i=1..length(s))))(""||(9*a(n-1))))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Apr 09 2015

a[n_] := a[n] = If[n==1, 1, IntegerReverse[9a[n-1]]];
a /@ Range[40] (* Jean-François Alcover, Jan 01 2021 *)

a(n) = reverse(9*a(n-1)) where a(1) = 1.

Conjecture: a(n)^(1/n) tends to 10. - Vaclav Kotesovec, Jan 03 2020

More terms from Alois P. Heinz, Apr 09 2015

A371880 Smallest number reachable starting from 1 and taking n steps either doubling or doubling+reversing.

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 46, 29, 58, 71, 34, 68, 37, 47, 49, 89, 79, 59, 19, 38, 67, 35, 7, 14, 28, 56, 13, 26, 25, 5, 1, 2, 4, 8, 16, 23, 46, 29, 58, 17, 34, 68, 37, 47, 49, 89, 79, 59, 19, 38, 67, 35, 7, 14, 28, 56, 13, 26, 25, 5, 1, 2, 4, 8, 16, 23, 46, 29, 58, 17

Offset: 0

Bryle Morga, Apr 14 2024

At each step you are allowed to either double x -> 2*x or double and reverse x -> R(2*x), where R(x) = A004086(x) is decimal digit reversal.
From Michael S. Branicky, Apr 14 2024: (Start)
Since a(30) = 1 = a(0), a(n) <= a(n-30) for n >= 30. a(39) <= 17 < a(9) = 71 is the first term that strictly lowers the bound. Is it eventually periodic?
Under the map, a term k has preimage k/2 if k is even plus terms of the form R(k)*10^i/2 for i > 1 and for i=0 if R(k) is even. (End)
The condition above implies a(i+30k) is nonincreasing for k >= 0 for all i in 0..29, hence it is periodic (with period being a factor of 30). When does the periodic part of the sequence begin? - Bryle Morga, Apr 15 2024
From David A. Corneth, Apr 15 2024: (Start)
a(n) == 2^n (mod 9).
Because of this, all values 1 <= a(n) <= 9 have a(n + 30*k) = a(n). That is a(30*k) = 1, a(30*k + 1) = 2, a(30*k + 2) = 4, a(30*k + 3) = 8, a(30*k + 22) = 7, a(30*k + 29) = 5, for k >= 0. (End)
Ultimately periodic sequence of period 30 with a(k+30)=a(k) for k != 9. - David Wilson, Apr 19 2024

			a(20) = 67 and here is the 20-move combination that reaches 67: 1, 2, 4, 8, 61, 221, 244, 884, 8671, 17342, 48643, 97286, 275491, 289055, 11875, 23750, 47500, 95000, 190000, 380000, 67.
a(21) = 35 and here is the 21-move combination that reaches 35: 1, 2, 4, 8, 61, 221, 244, 884, 8671, 17342, 48643, 97286, 275491, 289055, 11875, 23750, 47500, 95000, 91, 281, 265, 35.
a(30) = 1 using the path: 1, 2, 4, 8, 61, 122, 442, 488, 976, 2591, 2815, 365, 37, 47, 49, 98, 196, 392, 487, 479, 859, 1718, 3436, 2786, 2755, 155, 13, 26, 25, 5, 1. - _Michael S. Branicky_, Apr 14 2024

David A. Corneth, PARI program.
Bryle Morga et al., Optimal strategy in a game where Doubling and Doubling+Reversing are the allowed moves.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000079, A004086, A004094, A036447, A371966.

PARI
```
\\ See PARI link
```

def f(k, d):
      if d == 0:
            return k
      else:
            return min(f(2*k, d-1), f(int(str(2*k)[::-1]), d - 1))
def a(n):
      return f(1, n)
for n in range(25):
      print(a(n))

from itertools import islice
def reverse(n): return int(str(n)[::-1])
def agen(): # generator of terms
    reach = {1}
    while True:
        yield min(reach)
        newreach = set()
        for q in reach: newreach.update([2*q, reverse(2*q)])
        reach = newreach
print(list(islice(agen(), 28))) # Michael S. Branicky, Apr 14 2024

a(n) = f(1, n) where f(k, 0) = k and f(k, n) = min(f(2*k, n-1), f(R(2*k), n-1)).
a(30k) = 1 for k >= 0. - Michael S. Branicky, Apr 14 2024
a(9) = 71. For k != 9, a(k) is the minimum of the positive residues mod 99 of 2^k and 10*2^k. - David Wilson, Apr 19 2024

a(27)-a(34) from Michael S. Branicky, Apr 14 2024

a(35) and beyond from David Wilson, Apr 19 2024

cp's OEIS Frontend

A321539 3^n with digits rearranged into nonincreasing order.

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A321540 3^n with digits rearranged into nondecreasing order.

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A321541 a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nonincreasing order.

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A045539 Multiply by 5 and reverse.

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Maple

Mathematica

A132064 Numbers multiplied by 4 and written backwards.

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Maple

Mathematica

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A132078 Multiply previous term by 6 and reverse.

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Magma

Mathematica

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A132113 Multiply previous term by 8 and reverse.

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Maple

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A132114 Multiply previous term by 7 and reverse.

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