A023089 -id:A023089 - cp's OEIS frontend

A023086 Numbers k such that k and 2*k are anagrams.

0, 125874, 128574, 142587, 142857, 258714, 258741, 285714, 285741, 412587, 412857, 425871, 428571, 1025874, 1028574, 1042587, 1042857, 1052874, 1054287, 1072854, 1074285, 1078524, 1078542, 1085274, 1085427, 1087254, 1087425, 1087524, 1087542

Offset: 1

David W. Wilson

All terms are divisible by 9. - Eric M. Schmidt, Jul 12 2014
If x and y are in the sequence, then so is 10^m*x + y if y < 10^m. - Robert Israel, Mar 20 2017
From Petros Hadjicostas, Jul 29 2020: (Start)
This is Schuh's (1968) "doubles puzzle" (the double of k is 2*k). On five pages of his book, he finds the twelve 6-digit numbers that belong to this sequence (a(2) to a(13)) and the 288 7-digit numbers of the sequence (a(14) to a(301)).
All numbers in this sequence are permutations of numbers that are combinations of numbers from A336670, which is related to another puzzle of Schuh (1968). Before he solved this puzzle, he had to solve the puzzle described in A336670.
For example, a(2) = 125874 through a(13) = 428571 are all permutations of the number 512874, which is a combination of the numbers 512 and 874 that appear in A336670. (End)

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 31-35.

David W. Wilson, Table of n, a(n) for n = 1..10001
Mark Dominus, When do n and 2n have the same digits?

Cf. A023087, A023088, A023089, A023090, A023091, A023092, A023093, A336670.

Res:= 0:
for d from 1 to 7 do
  for n from 10^(d-1)+8 to 5*10^(d-1)-1 by 9 do
    if sort(convert(n,base,10)) = sort(convert(2*n,base,10)) then
      Res:= Res, n
    fi
od od:
Res; # Robert Israel, Mar 20 2017

si[n_] := Sort@ IntegerDigits@ n; Flatten@ {0, Table[ Select[ Range[ 10^e+8, 5*10^e-1, 9], si[#] == si[2 #] &], {e, 6}]} (* Giovanni Resta, Mar 20 2017 *)

def ok(n): return sorted(str(n)) == sorted(str(2*n))
print(list(filter(ok, range(1087543)))) # Michael S. Branicky, May 21 2021

# use with ok above for larger terms
def auptod(maxd):
  return [0] + list(filter(ok, (n for d in range(2, maxd+1) for n in range(10**(d-1)-1, 5*10**(d-1), 9))))
print(auptod(7)) # Michael S. Branicky, May 22 2021

A023087 Numbers k such that k and 3*k are anagrams.

Original entry on oeis.org

0, 1035, 2475, 10035, 10350, 12375, 14247, 14724, 23751, 24147, 24714, 24750, 24876, 24975, 27585, 28575, 100035, 100350, 102375, 103428, 103500, 107235, 113724, 114237, 123507, 123714, 123750, 123876, 123975, 124137, 128034, 134505, 135045

Offset: 1

David W. Wilson

All terms are divisible by 9. - Eric M. Schmidt, Jul 12 2014
From Petros Hadjicostas, Jul 28 2020: (Start)
This is Schuh's (1968) "treble puzzle" (the treble of k is 3*k). On five pages of his book, he finds the two 4-digit numbers that belong to this sequence (1035 and 2475), the thirteen 5-digit numbers of the sequence and the 104 6-digit numbers of the sequence. Note that if m belongs to the sequence, so does 10*m.
All numbers in this sequence are permutations of numbers that are combinations of numbers from A336661, which is related to another puzzle of Schuh (1968). Before he solved this puzzle, he had to solve the puzzle described in A336661.
For example, 1035 is a permutation of the number 3015 which is a combination of the numbers 301 and 5 that appear in A336661. As another example, note that 12375 and 23751 are both permutations of 31725, which is formed by combining the numbers 31, 72 and 5 from sequence A336661.
If we also admit zeros as initial digits, then we find more solutions to this sequence: 0351, 00351, 01035, 03501, 02475, ... These numbers are also permutations of numbers that can be formed by combining numbers in A336661. (End)

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 25-31.

Zak Seidov and David W. Wilson, Table of n, a(n) for n = 1..10001 (first 3000 terms from Zak Seidov)

Cf. A023086, A023088, A023089, A023090, A023091, A023092, A023093, A336661.

si[n_] := Sort@ IntegerDigits@ n; Flatten@ {0, Table[ Select[ Range[10^d + 8, 4 10^d - 1, 9], si[#] == si[3 #] &], {d, 0, 6}]} (* Giovanni Resta, Mar 20 2017, corrected by Philippe Guglielmetti, Jul 16 2018 *)

A023088 Numbers k such that k and 4*k are anagrams.

Original entry on oeis.org

0, 1782, 2178, 16782, 17604, 17802, 17820, 17832, 17982, 18027, 19728, 19782, 20178, 21678, 21780, 21783, 21798, 21978, 23958, 102564, 103845, 104769, 104895, 105264, 106254, 114528, 125406, 125604, 126054, 128052, 128205, 140256, 140526, 142857

Offset: 1

David W. Wilson

All terms are divisible by 3. - Eric M. Schmidt, Jul 12 2014

David W. Wilson, Table of n, a(n) for n = 1..10001

Cf. A023086, A023087, A023089, A023090, A023091, A023092, A023093.

si[n_] := Sort@ IntegerDigits@ n; Flatten@{0, Table[Select[Range[10^d + 2, 3 10^d - 1, 3], si[#] == si[4 #] &], {d, 0, 6}]} (* Giovanni Resta, Mar 20 2017, corrected by Philippe Guglielmetti, Jul 16 2018 *)
Select[Range[0,150000,3],Sort[IntegerDigits[#]]==Sort[ IntegerDigits[ 4#]]&] (* Harvey P. Dale, Dec 23 2017 *)

A023093 Numbers k such that k and 9*k are anagrams.

Original entry on oeis.org

0, 1089, 10089, 10449, 10890, 10899, 10989, 100089, 100449, 100890, 100899, 100989, 102249, 104490, 104499, 106749, 107793, 107892, 108900, 108990, 108999, 109890, 109899, 109989, 1000089, 1000449, 1000890, 1000899, 1000989, 1002249, 1004490

Offset: 1

David W. Wilson

All terms are divisible by 9. - Eric M. Schmidt, Jul 12 2014

David W. Wilson, Table of n, a(n) for n = 1..10001

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092.

si[n_] := Sort@ IntegerDigits@ n; Flatten@{0, Table[Select[Range[10^d + 8, 2 10^d - 1, 9], si[#] == si[9 #] &], {d, 0, 6}]} (* Giovanni Resta, Mar 20 2017 *)

A023090 Numbers k such that k and 6*k are anagrams.

Original entry on oeis.org

0, 1386, 13860, 13986, 14085, 14859, 15192, 106848, 120267, 126702, 126873, 137286, 138402, 138600, 138627, 139860, 139986, 140085, 140184, 140850, 140985, 141858, 142857, 148509, 148590, 148599, 149085, 149859, 150192, 151893, 151920, 151992

Offset: 1

David W. Wilson

All terms are divisible by 9. - Eric M. Schmidt, Jul 12 2014

David W. Wilson, Table of n, a(n) for n = 1..10001

Cf. A023086, A023087, A023088, A023089, A023091, A023092, A023093.

Select[Range[0,160000],Sort[IntegerDigits[#]]==Sort[IntegerDigits[ 6#]]&] (* Harvey P. Dale, Mar 18 2015 *)

A023091 Numbers k such that k and 7*k are anagrams.

Original entry on oeis.org

0, 1359, 11688, 11883, 12903, 13029, 13359, 13449, 13590, 13599, 13659, 107583, 108726, 111873, 116688, 116880, 116988, 118731, 118830, 118833, 119883, 123876, 124875, 126888, 127389, 129003, 129030, 129033, 129903, 130029, 130149, 130290

Offset: 1

David W. Wilson

All terms are divisible by 3. - Eric M. Schmidt, Jul 12 2014

David W. Wilson, Table of n, a(n) for n = 1..10001

Cf. A023086, A023087, A023088, A023089, A023090, A023092, A023093.

si[n_] := Sort@ IntegerDigits@ n; Flatten@{0, Table[Select[Range[10^d + 2, 2 10^d - 1, 3], si[#] == si[7 #] &], {d, 0, 6}]} (* Giovanni Resta, Mar 20 2017 *)
Select[3*Range[0,50000],Sort[IntegerDigits[7 #]]==Sort[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 06 2022 *)

A023092 Numbers k such that k and 8*k are anagrams.

Original entry on oeis.org

0, 113967, 116397, 1014138, 1045638, 1064538, 1139148, 1139670, 1139967, 1141398, 1156392, 1163970, 1163997, 10014138, 10045638, 10064538, 10141380, 10145628, 10148913, 10149138, 10182564, 10265382, 10268145, 10288836, 10289637, 10296387

Offset: 1

David W. Wilson

From Robert G. Wilson v, Oct 25 2012: (Start)
10^(k-1) < a(n) < 10^k/8 for all n > 0 and some k.
Number of terms < 10^k: 1, 1, 1, 1, 1, 3, 13, 92, 725, 5578, 41312, ...
First term > 10^k: 113967, 1014138, 10014138, 100014138, 1000014138, 10000014138, 100000014138, ...
First term < 10^k: 116397, 1163997, 12395169, 124839279, 1249839279, 12499839279, 124999839279, ...
(End)
All terms are divisible by 9. - Eric M. Schmidt, Jul 12 2014

David W. Wilson, Table of n, a(n) for n = 1..10001

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023093.

fQ[n_] := Sort@ IntegerDigits[ n] == Sort@ IntegerDigits[ 8n]; Select[ Range[0, 10328885], fQ] (* Robert G. Wilson v, Oct 25 2012 *)
Select[Range[0,103*10^5,9],Sort[IntegerDigits[#]]==Sort[IntegerDigits[8#]]&] (* Harvey P. Dale, Jul 18 2025 *)

A133220 a(n) is the smallest positive number k such that k and n*k are anagrams.

Original entry on oeis.org

1, 125874, 1035, 1782, 142857, 1386, 1359, 113967, 1089

Offset: 1

Tanya Khovanova, Oct 11 2007

			3*1035 = 3105 and this is the smallest such number.

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092, A023093.

Table[Select[Range[1000000], Sort[IntegerDigits[ # ]] == Sort[IntegerDigits[n# ]] &][[1]], {n, 9}]

A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

Original entry on oeis.org

1, 1035, 123876, 1402857, 1037520684, 142857

Offset: 1

Jean-Marc Rebert, Jun 04 2024

For n = 2..6 all terms are divisible by 9.
For n >= 4, a(n) must be divisible by 9, or a(n) = -1, because all anagrams d*k of k for d = 2, 3, 5, 6, 8 and 9 are divisible by 9. Thus there are only 3 values of d, i.e., 1, 4 and 7, for which k*d must not be divisible by 9.
If a(n) exists for n > 1 then 9|a(n). Holds for n = 2 and n = 3 by inspection. Proof for n >= 4: if k*d is an anagram of k where 2 <= d <= 9 then k*d - k = k*(d-1) is a multiple of 9. For this to be true, k must be a multiple of 9 as d is not of the form 1 (mod 3) for all d. - David A. Corneth, Jun 04 2024
From Michael S. Branicky, Jun 07 2024: (Start)
The following were constructed from multiples of cyclic numbers (cf. A180340, Wikipedia):
a(6)  = 142857 = (10^6 - 1) / 7;
a(7) <= 1304347826086956521739 = 3*(10^22 - 1) / 23;
a(8) <= 1176470588235294 = 2*(10^16 - 1) / 17;
a(9) <= 105263157894736842 = 2*(10^18 - 1) / 19. (End)

			a(2) = 1035, because 1035 * 1 = 1035 and 1035 * 3 = 3105 are anagrams of 1035, and no other number 1035 * i with digit i is an anagram of 1035, and no lesser number verifies this property.
Table n, k, set of multipliers.
  1   1          [1]
  2   1035       [1, 3]
  3   123876     [1, 3, 7]
  4   1402857    [1, 2, 3, 5]
  5   1037520684 [1, 2, 4, 5, 8]
  6   142857     [1, 2, 3, 4, 5, 6]

Wikipedia, Cyclic numbers.

Cf. A023086, A023087, A023088, A023089, A023090, A023091, A023092, A023093.
Cf. A245680, A245682.
Cf. A090061.
Cf. A180340, A001913, A036275.

isok(k, n) = my(d=vecsort(digits(k))); sum(i=1, 9, vecsort(digits(k*i)) == d) == n; \\ Michel Marcus, Jun 04 2024

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A023086 Numbers k such that k and 2*k are anagrams.

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A023087 Numbers k such that k and 3*k are anagrams.

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A023088 Numbers k such that k and 4*k are anagrams.

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A023093 Numbers k such that k and 9*k are anagrams.

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A023090 Numbers k such that k and 6*k are anagrams.

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A023091 Numbers k such that k and 7*k are anagrams.

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A023092 Numbers k such that k and 8*k are anagrams.

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A133220 a(n) is the smallest positive number k such that k and n*k are anagrams.

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A373407 Smallest positive integer k such that no more than n numbers (formed by multiplying k by a digit) are anagrams of k, or -1 if no such number exists.

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