A004186 -id:A004186 - cp's OEIS frontend

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

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

A052008 a(n) = 'n with digits sorted in ascending order' + 'n with digits sorted in descending order'.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88, 99, 110, 121

Offset: 0

Patrick De Geest, Nov 15 1999

a(n) = A004185(n) + A004186(n). - Reinhard Zumkeller, Jun 07 2015

			E.g., n = 19 -> 19 + 91 = 110.

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 0..9999 (First 1000 terms from Zak Seidov)
Nick Hobson, Python program for this sequence

Cf. A052009; different from A056964.

Cf. A004185, A004186, A271497.

a052008 n = a004185 n + a004186 n  -- Reinhard Zumkeller, Jun 07 2015

f[n_]:=Module[{sidn=Sort[IntegerDigits[n]]},FromDigits[sidn]+ FromDigits[ Reverse[sidn]]]; Array[f,70,0] (* Harvey P. Dale, Nov 13 2011 *)

for(n=0,100,D=digits(n);R=Vecrev(D);print1(sum(i=1,#D,10^(i-1)*(D[i]+R[i])),", ")) \\ Derek Orr, Feb 26 2017

A227362 Distinct digits of n arranged in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 21, 31, 41, 51, 61, 71, 81, 91, 20, 21, 2, 32, 42, 52, 62, 72, 82, 92, 30, 31, 32, 3, 43, 53, 63, 73, 83, 93, 40, 41, 42, 43, 4, 54, 64, 74, 84, 94, 50, 51, 52, 53, 54, 5, 65, 75, 85, 95, 60, 61, 62, 63, 64, 65, 6, 76, 86

Offset: 0

Reinhard Zumkeller, Jul 09 2013

a(n) <= 9876543210; a(a(n)) = a(n);
A055642(a(n)) <= 10;
A055642(a(n)) <= A055642(n), A055642(a(n)) = A055642(n) iff A178788(n) = 1;
a(A109303(n)) < A109303(n); a(A009995(n)) = A009995(n); a(A071589(n)) > A071589(n);
a(n) = A151949(n) + A180410(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004186, A230959.

import Data.List (nub, sort)
a227362 = read . reverse . sort . nub . show :: Integer -> Integer

a:= n-> parse(cat(sort([{convert(n, base, 10)[]}[]], `>`)[])):
seq(a(n), n=0..68);  # Alois P. Heinz, Sep 21 2022

f[n_] := FromDigits[Reverse@ Union@ IntegerDigits@ n]; f /@ Range[0, 68] (* Michael De Vlieger, Apr 16 2015, corrected by Robert G. Wilson v *)

a(n) = {if (n == 0, d = [0], d = digits(n)); eval(subst(Pol(vecsort(d,,12)), x, 10));} \\ Michel Marcus, Apr 16 2015

a(n)=fromdigits(vecsort(digits(n),,12)) \\ Charles R Greathouse IV, Apr 16 2015

def A227362(n): return int(''.join(sorted(set(str(n)),reverse=True))) # Chai Wah Wu, Nov 23 2022

A215014 Numbers where any two consecutive decimal digits differ by 1 after arranging the digits in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 102, 120, 123, 132, 201, 210, 213, 231, 234, 243, 312, 321, 324, 342, 345, 354, 423, 432, 435, 453, 456, 465, 534, 543, 546, 564, 567, 576, 645, 654, 657, 675, 678, 687, 756, 765, 768, 786, 789, 798, 867

Offset: 1

Tanya Khovanova and Charles R Greathouse IV, Jul 31 2012

a(4091131) = 9876543210 is the last term.
Numbers n such that A004186(n) is a term of A033075. - Felix Fröhlich, Dec 26 2017
Also 0 together with positive integers having k distinct digits and the difference between the largest and the smallest digit equal to k-1. - David A. Corneth, Dec 26 2017

Ely Golden, Table of n, a(n) for n = 1..10000

Cf. A004186, A033075.

lst = {}; Do[If[Times @@ Differences@Sort@IntegerDigits[n] == 1, AppendTo[lst, n]], {n, 0, 675}]; lst (* Arkadiusz Wesolowski, Aug 01 2012 *)
Join[Range[0,9],Select[Range[1000],Union[Differences[Sort[ IntegerDigits[ #]]]] == {1}&]] (* Harvey P. Dale, Jan 14 2015 *)

is(n)=my(v=vecsort(eval(Vec(Str(n)))));for(i=2,#v,if(v[i]!=1+v[i-1],return(0)));1

is(n) = if(!n, return(1)); my(d = digits(n), v = vecsort(d,,8)); #d == #v && v[#v] - v[1] == #v - 1

# Ely Golden, Dec 26 2017
def consecutive(li):
  for i in range(len(li)-1):
    if(li[i+1]!=1+li[i]): return False
  return True
def sorted_digits(n):
  lst=[]
  while(n>0):
    lst+=[n%10] ; n//=10
  lst.sort() ; return lst
j=0
for i in range(1,10001):
  while(not consecutive(sorted_digits(j))): j+=1
  print(str(i)+" "+str(j)) ; j+=1

# alternate for generating full sequence in seconds
from itertools import permutations as perms
frags = ["0123456789"[i:j] for i in range(10) for j in range(i+1, 11)]
afull = sorted(set(int("".join(s)) for f in frags for s in perms(f)))
print(afull[:70]) # Michael S. Branicky, Aug 04 2022

If zero is excluded, the number of terms with k digits, 1 <= k <= 10, is (11-k)*k! - (k-1)!. - Franklin T. Adams-Watters, Aug 01 2012

Name edited by Felix Fröhlich, Dec 26 2017

A319722 Write n in 5-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 16, 21, 10, 11, 12, 17, 22, 15, 16, 17, 18, 23, 20, 21, 22, 23, 24, 25, 30, 55, 80, 105, 30, 31, 56, 81, 106, 55, 56, 61, 86, 111, 80, 81, 86, 91, 116, 105, 106, 111, 116, 121, 50, 55, 60, 85, 110, 55, 56, 61, 86, 111, 60, 61, 62, 87, 112, 85

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..3124

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), this sequence (b=5), A319723 (b=6), A319724 (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165032, A319653.

Table[FromDigits[ReverseSort[IntegerDigits[n, 5]], 5], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 5), , 4), 5); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(5, 100)

n <= a(n) < 5n. - Charles R Greathouse IV, Aug 07 2024

A319723 Write n in 6-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 13, 19, 25, 31, 12, 13, 14, 20, 26, 32, 18, 19, 20, 21, 27, 33, 24, 25, 26, 27, 28, 34, 30, 31, 32, 33, 34, 35, 36, 42, 78, 114, 150, 186, 42, 43, 79, 115, 151, 187, 78, 79, 85, 121, 157, 193, 114, 115, 121, 127, 163, 199, 150, 151, 157, 163

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..7775

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), A319722 (b=5), this sequence (b=6), A319724 (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165051, A319654.

Table[FromDigits[ReverseSort[IntegerDigits[n, 6]], 6], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 6), , 4), 6); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(6, 100)

n <= a(n) < 6n. - Charles R Greathouse IV, Aug 07 2024

A328447 Smallest representative of the class of numbers having the same digits as n up to permutation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 12, 22, 23, 24, 25, 26, 27, 28, 29, 30, 13, 23, 33, 34, 35, 36, 37, 38, 39, 40, 14, 24, 34, 44, 45, 46, 47, 48, 49, 50, 15, 25, 35, 45, 55, 56, 57, 58, 59, 60, 16, 26, 36, 46, 56, 66, 67, 68, 69, 70, 17, 27, 37, 47, 57, 67

Offset: 0

M. F. Hasler, Oct 15 2019

Sort the digits in increasing order. If the list starts with a digit 0, move the smallest nonzero digit to the front.

Every term is in A179239. - David A. Corneth, Oct 17 2019

			a(201) = 102: largest digits go to the end, but the smallest nonzero digit must go first.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A179239, A004186 (largest representative of the class of n).

f:= proc(n) local L,i,t;
  L:= sort(convert(n,base,10));
  if L[1]=0 then
    t:= numboccur(0,L)+1;
    L:= [L[t],op(L[1..t-1]),op(L[t+1..-1])];
  fi;
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f(0):= 0:
map(f, [$0..100]);

Array[FromDigits@ If[First@ # == 0, Flatten@ MapAt[Reverse, TakeDrop[#, 2], 1], #] &@ Sort@ IntegerDigits[#] &, 67] (* Michael De Vlieger, Oct 17 2019 *)

A328447(n)={if(n=vecsort(digits(n)), n[1]|| for(k=2,#n,n[k]&&[n[1]=n[k],n[k]=0,break]));fromdigits(n)}

def A328447(n):
    if n == 0: return 0
    s = str(n)
    l, s = len(s), ''.join(sorted(s.replace('0','')))
    return int(s[0]+'0'*(l-len(s))+s[1:]) # Chai Wah Wu, Dec 06 2021

A319651 Largest number having in its ternary representation the same number of 0's, 1's and 2's as n.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 7, 8, 9, 12, 21, 12, 13, 22, 21, 22, 25, 18, 21, 24, 21, 22, 25, 24, 25, 26, 27, 36, 63, 36, 39, 66, 63, 66, 75, 36, 39, 66, 39, 40, 67, 66, 67, 76, 63, 66, 75, 66, 67, 76, 75, 76, 79, 54, 63, 72, 63, 66, 75, 72, 75, 78, 63, 66, 75, 66, 67, 76, 75, 76

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..6560

Base b: A073138 (b=2), this sequence (b=3), A319720 (b=4), A319722 (b=5), A319723 (b=6), A319724 (b=7), this sequence (b=8), A319726 (b=9), A004186 (b=10).

Cf. A038574.

Table[FromDigits[ReverseSort[IntegerDigits[n, 3]], 3], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 3),,4), 3); \\ Michel Marcus, Sep 25 2018

from gmpy2 import digits
def A319651(n):
    return int(''.join(sorted(digits(n,3),reverse=True)),3) # Chai Wah Wu, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(3, 100)

n <= a(n) < 3n. - Charles R Greathouse IV, Aug 07 2024

A319724 Write n in 7-ary, sort digits into decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 15, 22, 29, 36, 43, 14, 15, 16, 23, 30, 37, 44, 21, 22, 23, 24, 31, 38, 45, 28, 29, 30, 31, 32, 39, 46, 35, 36, 37, 38, 39, 40, 47, 42, 43, 44, 45, 46, 47, 48, 49, 56, 105, 154, 203, 252, 301, 56, 57, 106, 155, 204, 253, 302, 105, 106, 113, 162

Offset: 0

Seiichi Manyama, Sep 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..2400

b-ary: A073138 (b=2), A319651 (b=3), A319720 (b=4), A319722 (b=5), A319723 (b=6), this sequence (b=7), A319725 (b=8), A319726 (b=9), A004186 (b=10).

Cf. A165071, A319655.

Table[FromDigits[ReverseSort[IntegerDigits[n, 7]], 7], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 7), , 4), 7); \\ Michel Marcus, Sep 26 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.reverse.join.to_i(k)}
end
p A(7, 100)

n <= a(n) < 7n. - Charles R Greathouse IV, Aug 07 2024

cp's OEIS Frontend

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

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A321539 3^n with digits rearranged into nonincreasing order.

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A052008 a(n) = 'n with digits sorted in ascending order' + 'n with digits sorted in descending order'.

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A227362 Distinct digits of n arranged in decreasing order.

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A215014 Numbers where any two consecutive decimal digits differ by 1 after arranging the digits in decreasing order.

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A319722 Write n in 5-ary, sort digits into decreasing order.

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A319723 Write n in 6-ary, sort digits into decreasing order.

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