A004094 -id:A004094 - cp's OEIS frontend

A163632 Triple and reverse digits.

1, 3, 9, 72, 612, 6381, 34191, 375201, 3065211, 3365919, 75779001, 300733722, 661102209, 7266033891, 37610189712, 631965038211, 3364115985981, 34975974329001, 300789229729401, 302881986763209, 726982069546809

Offset: 1

Dmitry Kamenetsky, Aug 02 2009

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

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. A132064, A045539, A132078, A132114, A132113, A133361.

a[n_] := a[n] = If[n==1, 1, IntegerReverse[3a[n-1]]];
Array[a, 40] (* Jean-François Alcover, Jan 01 2021 *)

Offset changed from 0 to 1 by Vaclav Kotesovec, Jan 03 2020

A004167 Powers of 3 written backwards.

Original entry on oeis.org

1, 3, 9, 72, 18, 342, 927, 7812, 1656, 38691, 94095, 741771, 144135, 3234951, 9692874, 70984341, 12764034, 361041921, 984024783, 7641622611, 1044876843, 30235306401, 90695018313, 72887134149, 184635924282, 344906882748, 9238285681452, 7894847955267

Offset: 0

N. J. A. Sloane

From a(2) onwards, all terms are divisible by 9. - Alonso del Arte, Apr 04 2014

			a(5) = 342 since 3^5 = 243.

Alois P. Heinz, 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).

a:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||(3^n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 09 2015

Table[FromDigits[Reverse[IntegerDigits[3^n]]], {n, 0, 26}] (* Alonso del Arte, Apr 04 2014 *)

def A004167(n):
    return int(str(3**n)[::-1]) # Chai Wah Wu, Feb 19 2021

a(n) = A004086(A000244(n)). - Michel Marcus, Apr 05 2014

More terms from Eric M. Schmidt, Apr 04 2014

A057615 ATS: Add Then Sort (i.e., double previous term and then sort digits).

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 46, 29, 58, 116, 223, 446, 289, 578, 1156, 1223, 2446, 2489, 4789, 5789, 11578, 12356, 12247, 24449, 48889, 77789, 155578, 111356, 122227, 244445, 48889, 77789, 155578, 111356, 122227, 244445, 48889, 77789, 155578, 111356

Offset: 1

Henry Bottomley, Oct 09 2000

Starting from a(1)=1 sequence cycles starting from a(25) = 48889, 77789, 155578, 111356, 122227, 244445, 48889, ... etc.

			a(8)=29 since a(7)=46, 46 + 46 = 92 and 92 sorted is 29.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A033861 for STA, A004000 for RATS.

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[Sort[IntegerDigits[2#]]]&,1,40] (* Harvey P. Dale, Oct 03 2011 *)

from itertools import accumulate
def ats(anm1, _): return int("".join(sorted(str(2*anm1))))
print(list(accumulate([1]*40, ats))) # Michael S. Branicky, Jul 17 2021

G.f.: x*(-219996*x^29 - 109980*x^28 - 99000*x^27 - 144000*x^26 - 72000*x^25 - 44100*x^24 - 21960*x^23 - 9801*x^22 - 11133*x^21 - 10422*x^20 - 5211*x^19 - 4500*x^18 - 2043*x^17 - 2223*x^16 - 1107*x^15 - 1098*x^14 - 549*x^13 - 243*x^12 - 423*x^11 - 207*x^10 - 108*x^9 - 54*x^8 - 27*x^7 - 45*x^6 - 23*x^5 - 16*x^4 - 8*x^3 - 4*x^2 - 2*x - 1)/(x^6 - 1). - Chai Wah Wu, Nov 20 2018

A057708 Numbers m such that 2^m reversed is prime.

Original entry on oeis.org

1, 4, 5, 7, 10, 17, 24, 37, 45, 55, 70, 77, 107, 137, 150, 271, 364, 1157, 1656, 2004, 2126, 3033, 3489, 3645, 4336, 6597, 7279, 12690, 13840, 20108, 21693, 28888, 84155, 102930

Offset: 1

G. L. Honaker, Jr., Oct 23 2000

a(35) > 105000. - Giovanni Resta, Feb 22 2013
From Bernard Schott, Jan 30 2022: (Start)
If m is an even term, then u = m/2 is a term of A350441, this because 2^m = 4^(m/2). In fact, terms of A350441 are half the even terms of this sequence here.
If m is a term multiple of 3, then k = m/3 is a term of A350442, this because 2^m = 8^(m/3). First examples: m = 24, 45, 150, 1656, ... and corresponding k = 8, 15, 50, 552, ... (End)
a(35) > 200000. - Michael S. Branicky, May 12 2025

			4 is a term because 2^4 reversed is 61 and prime.

Cf. A004094, A341713.

Numbers m such that k^m reversed is prime: this sequence (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), A058995 (k=13).

with(numtheory): myarray := []: for n from 1 to 4000 do it1 := convert(2^n, base, 10): it2 := sum(10^(nops(it1)-i)*it1[i], i=1..nops(it1)): if isprime(it2) then printf(`%d,`,n) fi: od:

Do[ If[ PrimeQ[ FromDigits[ Reverse[ IntegerDigits[2^n]] ]], Print[ n]], {n, 20000}] (* Robert G. Wilson v, Jan 29 2005 *)
Select[Range[4400],PrimeQ[IntegerReverse[2^#]]&] (* Requires Mathematica version 10 or later *) (* The program generates the first 25 terms of the sequence; to generate more, increase the Range constant, but the program will take longer to run. *) (* Harvey P. Dale, Aug 05 2020 *)

isok(m) = isprime(fromdigits(Vecrev(digits(2^m)))) \\ Mohammed Yaseen, Jul 20 2022

from sympy import isprime
k, m, A057708_list = 1, 2,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A057708_list.append(k)
    k += 1
    m *= 2 # Chai Wah Wu, Mar 09 2021

More terms from Chris Nash (chris_nash(AT)prodigy.net), Oct 25 2000
Two more terms from Robert G. Wilson v, Jan 29 2001
3 more terms from Farideh Firoozbakht, Aug 05 2004
a(33)-a(34) from Giovanni Resta, Feb 22 2013

A321542 a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nondecreasing order.

Original entry on oeis.org

1, 3, 9, 27, 18, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135, 45, 135

Offset: 0

N. J. A. Sloane, Nov 19 2018

For n >= 5, alternates between 45 and 135.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1).

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).

A321542list[nmax_]:=PadRight[{1,3,9,27,18},nmax+1,{135,45}];A321542list[100] (* Paolo Xausa, Aug 10 2023 *)

G.f.: (-117*x^6 - 18*x^5 - 9*x^4 - 24*x^3 - 8*x^2 - 3*x - 1)/(x^2 - 1). - Chai Wah Wu, Nov 20 2018

A028909 Arrange digits of 2^n in ascending order.

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 46, 128, 256, 125, 124, 248, 469, 1289, 13468, 23678, 35566, 11237, 122446, 224588, 145678, 122579, 134449, 368888, 11266777, 23334455, 1466788, 112234778, 234455668, 12356789, 112344778, 1234446788, 2244667999

Offset: 0

N. J. A. Sloane

Leading zeros are discarded (e.g., 2^23 = 8388608 -> 0368888 becomes 368888).

Alois P. Heinz, Table of n, a(n) for n = 0..3600

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).

[Seqint(Reverse(Sort(Intseq(2^n)))):n in [0..35]]; // Vincenzo Librandi, Jan 22 2020

a:= n-> parse(cat(sort(convert(2^n, base, 10))[])):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2020

Table[FromDigits[Sort[IntegerDigits[2^n]]],{n,0,40}] (* Harvey P. Dale, Aug 20 2013 *)

def A028909(n):
    return int(''.join(sorted(str(2**n)))) # Chai Wah Wu, Feb 19 2021

More terms from Patrick De Geest, Apr 1998

A028910 Arrange digits of 2^n in descending order.

Original entry on oeis.org

1, 2, 4, 8, 61, 32, 64, 821, 652, 521, 4210, 8420, 9640, 9821, 86431, 87632, 66553, 732110, 644221, 885422, 8765410, 9752210, 9444310, 8888630, 77766211, 55443332, 88766410, 877432211, 866554432, 987653210, 8774432110, 8876444321

Offset: 0

N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 0..3321

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).

[Seqint(Sort(Intseq(2^n))):n in [0..31]]; // Marius A. Burtea, Oct 06 2019

a:= n-> parse(cat(sort(convert(2^n, base, 10), `>`)[])):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2020

FromDigits[Reverse[Sort[IntegerDigits[#]]]]&/@(2^Range[0,40]) (* Harvey P. Dale, Mar 06 2020 *)

def A028910(n):
    return int(''.join(sorted(str(2**n),reverse=True))) # Chai Wah Wu, Feb 19 2021

More terms from Patrick De Geest, Apr 15 1998

A263451 a(n) is the largest anagram of 2*a(n-1), a(1)=1.

Original entry on oeis.org

1, 2, 4, 8, 61, 221, 442, 884, 8761, 75221, 544210, 8842100, 87642100, 875422100, 8754421000, 88754210000, 877542100000, 8755421000000, 87542110000000, 875422100000000, 8754421000000000, 88754210000000000, 877542100000000000, 8755421000000000000

Offset: 1

Zak Seidov, Oct 18 2015

For large n, a(n)/a(n-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).

a263451 n = a263451_list !! (n-1)
a263451_list = iterate (a004186 . (* 2)) 1
-- Reinhard Zumkeller, Oct 19 2015

[n eq 1 select 1 else Seqint(Sort(Intseq(2*Self(n-1)))): n in [1..30]]; // Bruno Berselli, Oct 19 2015

s={1,2,4,8}; a=8; Do[b=FromDigits[Reverse[Sort[IntegerDigits[2*a]]]]; AppendTo[s,a=b],{20}]; s
NestList[FromDigits[ReverseSort[IntegerDigits[2 #]]]&,1,30] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, May 17 2019 *)

a(n) >= A036447(n).
From Alois P. Heinz, Oct 19 2015: (Start)
G.f.: x*(99990000000*x^18 +86679000000*x^17 -333332100000*x^16 -13533210000*x^15 +6579000*x^14 +8577900*x^13 +354357900*x^12 +212157900*x^11 +60455790*x^10 +7924779*x^9 +3991239*x^8 +1999116*x^7 +999558*x^6 -221*x^5 -61*x^4 -8*x^3 -4*x^2 -2*x -1) / ((10*x-1) *(1+10*x) *(100*x^2+10*x+1) *(100*x^2-10*x+1)).
a(n) = 10^6 * a(n-6) for n >= 20. (End)
a(n+1) = A004186(2*a(n)). - Reinhard Zumkeller, Oct 19 2015

A321539 3^n with digits rearranged into nonincreasing order.

Original entry on oeis.org

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

cp's OEIS Frontend

A163632 Triple and reverse digits.

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A004167 Powers of 3 written backwards.

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A057615 ATS: Add Then Sort (i.e., double previous term and then sort digits).

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A057708 Numbers m such that 2^m reversed is prime.

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A321542 a(0)=1; thereafter a(n) = 3*a(n-1) with digits rearranged into nondecreasing order.

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A028909 Arrange digits of 2^n in ascending order.

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A028910 Arrange digits of 2^n in descending order.

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