A004185 -id:A004185 - cp's OEIS frontend

A319655 Write n in 7-ary, sort digits into increasing order.

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 9, 16, 17, 18, 19, 20, 3, 10, 17, 24, 25, 26, 27, 4, 11, 18, 25, 32, 33, 34, 5, 12, 19, 26, 33, 40, 41, 6, 13, 20, 27, 34, 41, 48, 1, 8, 9, 10, 11, 12, 13, 8, 57, 58, 59, 60, 61, 62, 9, 58, 65, 66, 67, 68, 69, 10, 59, 66, 73

Offset: 0

Seiichi Manyama, Sep 25 2018

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

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), this sequence (b=7), A319656 (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165071, A319724.

a:= n-> (l-> add(l[-i]*7^(i-1), i=1..nops(l)))(sort(convert(n, base, 7))):
seq(a(n), n=0..73);  # Alois P. Heinz, Aug 07 2024

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

a(n) = fromdigits(vecsort(digits(n, 7)), 7); \\ Michel Marcus, Sep 25 2018

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

A319656 Write n in 8-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 2, 10, 18, 19, 20, 21, 22, 23, 3, 11, 19, 27, 28, 29, 30, 31, 4, 12, 20, 28, 36, 37, 38, 39, 5, 13, 21, 29, 37, 45, 46, 47, 6, 14, 22, 30, 38, 46, 54, 55, 7, 15, 23, 31, 39, 47, 55, 63, 1, 9, 10, 11, 12, 13, 14

Offset: 0

Seiichi Manyama, Sep 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..4095

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), this sequence (b=8), A319657 (b=9), A004185 (b=10).

Cf. A165090.

a:= n-> (l-> add(l[-i]*8^(i-1), i=1..nops(l)))(sort(convert(n, base, 8))):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 8]], 8], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 8)), 8); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(8, 100)

A319657 Write n in 9-ary, sort digits into increasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 10, 11, 12, 13, 14, 15, 16, 17, 2, 11, 20, 21, 22, 23, 24, 25, 26, 3, 12, 21, 30, 31, 32, 33, 34, 35, 4, 13, 22, 31, 40, 41, 42, 43, 44, 5, 14, 23, 32, 41, 50, 51, 52, 53, 6, 15, 24, 33, 42, 51, 60, 61, 62, 7, 16, 25, 34, 43, 52, 61, 70, 71

Offset: 0

Seiichi Manyama, Sep 25 2018

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

b-ary: A038573 (b=2), A038574 (b=3), A319652 (b=4), A319653 (b=5), A319654 (b=6), A319655 (b=7), A319656 (b=8), this sequence (b=9), A004185 (b=10).

Cf. A165110.

a:= n-> (l-> add(l[-i]*9^(i-1), i=1..nops(l)))(sort(convert(n, base, 9))):
seq(a(n), n=0..71);  # Alois P. Heinz, Aug 07 2024

Table[FromDigits[Sort[IntegerDigits[n, 9]], 9], {n, 0, 100}] (* Paolo Xausa, Aug 07 2024 *)

a(n) = fromdigits(vecsort(digits(n, 9)), 9); \\ Michel Marcus, Sep 25 2018

def A(k, n)
  (0..n).map{|i| i.to_s(k).split('').sort.join.to_i(k)}
end
p A(9, 100)

A211654 Primes that remain prime when their digits are sorted into nondecreasing order.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 47, 59, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 173, 179, 193, 197, 199, 223, 227, 229, 233, 239, 257, 269, 271, 277, 293, 307, 311, 317, 337, 347, 349, 359, 367, 373

Offset: 1

Francis J. McDonnell, Apr 17 2012

In sequence A004185 these are referred to as "sortable primes". Nontrivial terms (with digits not in nondecreasing order) are listed in A086042. - M. F. Hasler, Jul 30 2019.

			173 is prime and after the digits are sorted into nondecreasing order we obtain 137, which is prime.

Francis J. McDonnell, Table of n, a(n) for n = 1..10000
Francis J. McDonnell, Java Program

Cf. A028864, A028867, A211655.

Cf. A086042 (nontrivial solutions), A004185 (n with digits sorted).

[p:p in PrimesUpTo(400)| IsPrime(Seqint(Reverse(Sort(Intseq(p,10)))))]; // Marius A. Burtea, Jul 30 2019

Select[Prime[Range[200]], PrimeQ[FromDigits[Sort[IntegerDigits[#]]]] &] (* T. D. Noe, Apr 17 2012 *)

select( is_A211654(p)={isprime(fromdigits(vecsort(digits(p))))&&isprime(p)}, primes([1,999])) \\ M. F. Hasler, Jul 30 2019

A028905 Arrange digits of primes in ascending order.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 13, 37, 14, 34, 47, 35, 59, 16, 67, 17, 37, 79, 38, 89, 79, 11, 13, 17, 19, 113, 127, 113, 137, 139, 149, 115, 157, 136, 167, 137, 179, 118, 119, 139, 179, 199, 112, 223, 227, 229, 233, 239, 124, 125, 257, 236, 269

Offset: 1

N. J. A. Sloane

Leading zeros are discarded (e.g., 107, rearranged to 017, becomes 17).

			The digits of 41 are 4, 1, which sorted are 1, 4; those are reinterpreted as 14.
The digits of 43 are 4, 3, which sorted are 3, 4; those are reinterpreted as 34.
The digits of 47 are 4, 7, which are already sorted, so 47 is not changed.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A028906, A028864.

Cf. A004185, A000040.

a028905 = a004185 . a000040  -- Reinhard Zumkeller, Apr 03 2015

Table[FromDigits[Sort[IntegerDigits[Prime[n]]]], {n, 100}] (* Alonso del Arte, Nov 25 2019 *)

eva(n) = subst(Pol(n), x, 10)
a(n) = eva(vecsort(digits(prime(n)))) \\ Felix Fröhlich, Nov 25 2019

a(n) = A004185(A000040(n)). - Reinhard Zumkeller, Apr 03 2015

a(n) = prime(n) if prime(n) is in A028864. - Alonso del Arte, Nov 25 2019

More terms from Patrick De Geest, Apr 1998

Offset corrected by Reinhard Zumkeller, Apr 03 2015

A065641 Smallest number with persistence n for the sort-and-subtract-sequence.

Original entry on oeis.org

0, 1, 10, 60, 90, 101, 120, 380, 450, 505, 807, 1020, 1070, 1303, 1450, 3810, 10020, 10404, 10560, 16056, 16200, 18088, 20322, 20580, 35790, 79000, 80088, 90877, 243700, 279509, 330832, 374330, 380038, 903655, 1002404, 1005064, 1020828

Offset: 0

Ulrich Schimke (ulrschimke(AT)aol.com), Dec 03 2001

Sort the digits of an integer and subtract the result from the original. Continue with the result until you reach 0. The sequence gives the least integer that needs n steps to reach 0.

			60 is the smallest number that needs 3 steps to reach 0: 60 -> 60 - 06 = 54 -> 54 - 45 = 9 -> 9 - 9 = 0, hence a(3) = 60.

David W. Wilson and Reinhard Zumkeller, Table of n, a(n) for n = 0..100 (a(n) for n = 1..55 from Reinhard Zumkeller).

Cf. A004185, A193582.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a065641 n = a065641_list !! (n-1)
a065641_list = map (fromJust . (`elemIndex` a193582_list)) [1..]
-- Reinhard Zumkeller,Aug 10 2011

Persist[n_] := Length[NestWhileList[# - FromDigits[Sort[IntegerDigits[#]]] &, n, # != 0 &]] - 1; nn = 20; t = Table[0, {nn}]; cnt = 0; k = 0; While[cnt < nn, k++; c = Persist[k]; If[c <= nn && t[[c]] == 0, t[[c]] = k; cnt++]]; t  (* Harvey P. Dale, Mar 24 2011 *)
persist[n_]:=Length[NestWhileList[#-FromDigits[Sort[IntegerDigits[#]]]&,n,#!=0&]]-1; Module[ {nn=103*10^4,tbl},tbl=Table[{n,persist[n]},{n,0,nn}];DeleteDuplicates[ tbl,GreaterEqual[ #1[[2]],#2[[2]]]&]][[;;,1]] (* Harvey P. Dale, Sep 03 2023 *)

Added a(0) = 0. David W. Wilson, Jan 08 2017

A193582 Persistence of n for sort-and-subtract (A193581).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2

Offset: 0

Reinhard Zumkeller, Aug 10 2011

Number of sort-and-subtract steps to reach 0 when starting with n.

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

Cf. A004185.

a193582 n = length $ takeWhile (> 0) $ iterate a193581 n
a193582_list = map a193582 [0..]

A221714 Numbers written in base 2 with digits rearranged to be in decreasing order.

Original entry on oeis.org

0, 1, 10, 11, 100, 110, 110, 111, 1000, 1100, 1100, 1110, 1100, 1110, 1110, 1111, 10000, 11000, 11000, 11100, 11000, 11100, 11100, 11110, 11000, 11100, 11100, 11110, 11100, 11110, 11110, 11111, 100000, 110000, 110000, 111000, 110000

Offset: 0

Bruce L. Rothschild and N. J. A. Sloane, Jan 26 2013

This is the base-2 equivalent of A004186.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

For decimal equivalents see A073138.

Cf. A004086, A004185, A004186, A009996.

a:= n-> parse(cat(0, sort(Bits[Split](n), `>`)[])):
seq(a(n), n=0..36);  # Alois P. Heinz, Aug 18 2025

a[n_] := FromDigits[-Sort[-IntegerDigits[n, 2]]] (* Giovanni Resta, Jan 27 2013 *)

def a(n):
     return "".join(sorted(bin(n)[2:],reverse=True)) # Indranil Ghosh, Jan 09 2017

def A221714(n): return int(bin((m:=1<>n.bit_count()))[2:]) # Chai Wah Wu, Aug 18 2025

a(18)-a(36) from Giovanni Resta, Jan 27 2013

A237568 Fibonacci-like sequence of numbers with nondecreasing positive digits. Let a^+ denote the number that is obtained from a if its positive digits are written in nondecreasing order, while zeros remain in their places. Let a<+>b = (a + b)^+. a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 12, 25, 37, 26, 36, 26, 26, 25, 15, 40, 55, 59, 114, 137, 125, 226, 135, 136, 127, 236, 336, 257, 359, 166, 255, 124, 379, 305, 468, 377, 458, 358, 168, 256, 244, 500, 447, 479, 269, 478, 477, 559, 1036, 1559, 2559, 1148, 3707, 4558, 2568, 1267, 3358, 2456, 1458, 1349, 2708, 4057, 5667, 2479, 1468, 3479, 4479, 5789, 10268, 15067, 23355, 22348

Offset: 0

Vladimir Shevelev, Feb 09 2014

Note that operation n^+ differs from the one in A004185. If a term of the sequence has k digits, then it is followed by terms with >=k digits. The sequence has 7 terms with 1 digit, 13 terms with 2 digits, 30 terms with 3 digits, etc. The corresponding maximal terms are 8, 59, 559, etc.

The sequence is eventually periodic with period of length 144 and the first position of period 237. - Peter J. C. Moses, Feb 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 0..952

Cf. A000045, A001129, A004185, A069638.

a[0]:=0;a[1]:=1;a[n_]:=a[n]=FromDigits[Insert[DeleteCases[Sort[#],0],0,1+#-Range[Length[#]]&[Position[#,0]]]&[IntegerDigits[a[n-1]+a[n-2]]]]; Map[a,Range[0,99]] (* Peter J. C. Moses, Feb 09 2014 *)

A237575 Fibonacci-like numbers with nonincreasing positive digits. Let a denote the number that is obtained from a if its digits are written in nonincreasing order. Let a<+>b = (a + b). a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 31, 93, 421, 541, 962, 5310, 7622, 93221, 843100, 963321, 8642110, 9654310, 98642210, 986522100, 8654311100, 9864332000, 88654311100, 98865431100, 987754221100, 9866652211000, 86544432110000, 98644321110000, 888755322110000

Offset: 0

Vladimir Shevelev, Feb 09 2014

Peter J. C. Moses, Table of n, a(n) for n = 0..499

Cf. A000045, A001129, A004185, A069638, A237568.

a:= proc(n) option remember; `if`(n<2, n, parse(cat(
      sort(convert(a(n-1)+a(n-2), base, 10), `>`)[])))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 31 2022

a[0]:=0;a[1]:=1;a[n_]:=a[n]=FromDigits[Reverse[Sort[IntegerDigits[a[n-1]+a[n-2]]]]];Map[a,Range[0,20]] (* Peter J. C. Moses, Feb 09 2014 *)

Correction and extension by Peter J. C. Moses

cp's OEIS Frontend

A319655 Write n in 7-ary, sort digits into increasing order.

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Maple

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A319656 Write n in 8-ary, sort digits into increasing order.

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Maple

Mathematica

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Ruby

A319657 Write n in 9-ary, sort digits into increasing order.

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Maple

Mathematica

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Ruby

A211654 Primes that remain prime when their digits are sorted into nondecreasing order.

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Magma

Mathematica

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A028905 Arrange digits of primes in ascending order.

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Haskell

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A065641 Smallest number with persistence n for the sort-and-subtract-sequence.

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Haskell

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A193582 Persistence of n for sort-and-subtract (A193581).

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A237575 Fibonacci-like numbers with nonincreasing positive digits. Let a denote the number that is obtained from a if its digits are written in nonincreasing order. Let a<+>b = (a + b). a(0)=0, a(1)=1, for n>=2, a(n) = a(n-1) <+> a(n-2).