A023087 -id:A023087 - 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

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 *)

A023089 Numbers k such that k and 5*k are anagrams.

Original entry on oeis.org

0, 142857, 148257, 174285, 174825, 1025748, 1028574, 1057428, 1057482, 1082574, 1085742, 1402857, 1408257, 1420857, 1425708, 1425789, 1425897, 1428057, 1428570, 1428579, 1428597, 1429785, 1429857, 1457028, 1457082, 1457829, 1458297, 1480257

Offset: 1

David W. Wilson

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

This is Schuh's (1968) "quintuples puzzle". - Petros Hadjicostas, Jul 28 2020

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 35-37.

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

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

si[n_] := Sort@ IntegerDigits@ n; Flatten@{0, Table[Select[Range[10^d + 8, 2 10^d - 1, 9], si[#] == si[5 #] &], {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 *)

A336661 Numbers that have decimal expansion c(1)c(2)...c(n) with distinct digits that satisfy c(1) <> 0, c(1) is the largest digit, and for each i in 1..n there is j in 0..2 such that c(i) == 3*c(i-1) + j (mod 10) (with c(0): = c(n)).

Original entry on oeis.org

0, 4, 5, 9, 31, 72, 86, 301, 431, 602, 715, 842, 856, 973, 986, 4301, 6015, 7142, 7302, 7315, 8426, 8572, 8602, 9713, 9726, 9843, 9856, 60142, 71302, 73015, 73142, 84302, 85602, 85726, 97143, 97156, 97286, 98426, 98573, 714302, 715602, 726015, 730142, 843026, 843156, 857142, 857302, 860142, 971426, 972843, 972856, 973026, 973156, 984273, 985713, 985726, 986013

Offset: 1

Petros Hadjicostas, Jul 28 2020

This is one of Schuh's examples of a puzzle tree.
Putting the number on a circle and going clockwise, we observe that a 0 is followed by a 1 or 2; a 1 is followed by a 3, 4, or 5; a 2 is followed by a 6, 7, or 8; a 3 is followed by a 0, 1, or 9; a 4 is followed by a 2 or 3; a 5 is followed by a 6 or 7; a 6 is followed by a 0, 8, or 9; a 7 is followed by a 1, 2, or 3; an 8 is followed by a 4, 5, or 6; and a 9 is followed by a 7 or 8. (These observations assume the number has at least two digits.)
Schuh (pp. 25-31) uses the solution to this problem to solve the "trebles puzzle": find all numbers (with no initial 0) that are written with the same digits as their treble (the treble of k is 3*k). These numbers are listed in A023087.
The number 0 has been included here for two reasons: (i) we may assume that it satisfies the conditions of the problem vacuously, and (ii) its inclusion allows Schuh to solve the "treble puzzle". The numbers in A023087 are all permutations of combinations of numbers in this sequence.

			In all the cases below, the first digit must be the largest and all the digits must be distinct.
4 belongs to this list because c(1) = 4 = c(0) and 4 == 3*4 + 2 (mod 10).
31 belongs to this list because c(1) = 3, c(2) = 1 = c(0), 3 == 3*1 (mod 10), and 1 == 3*3 + 2 (mod 10).
301 belongs to this list because 3 == 3*1 (mod 10), 0 == 3*3 + 1 (mod 10), and 1 == 3*0 + 1 (mod 10).
4301 belongs to this list because 4 == 3*1 + 1 (mod 10), 3 == 3*4 + 1 (mod 10), 0 == 3*3 + 1 (mod 10), and 1 == 3*0 + 1 (mod 10).
60142 belongs to this list because 6 == 3*2 (mod 10), 0 == 3*6 + 2 (mod 10), 1 == 3*0 + 1 (mod 10), 4 = 3*1 + 1 (mod 10), and 2 = 3*4 (mod 10).

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968, pp. 8-10.

Petros Hadjicostas, Table of n, a(n) for n = 1..141
David A. Corneth, PARI program. [It generates all 141 terms of this finite sequence.]
Wikipedia, Frederik Schuh.
Dutch Wikipedia, Frederik Schuh. [Has more extensive biography in Dutch.]

Cf. A023087.

PARI
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See Corneth link
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A159837 Numbers n such that n, 3n and 9n are anagrams.

Original entry on oeis.org

10348965, 100348965, 103489650, 103489965, 103498965, 1000348965, 1003489650, 1003489965, 1003498965, 1023748965, 1032890967, 1032896709, 1032908967, 1032967089, 1034896500, 1034897625, 1034899650, 1034899965, 1034989650

Offset: 1

Zak Seidov, Apr 23 2009

Numbers n such that n and 3n are terms in A023087.
Intersection of A023087 and A023093.
Sequence is infinite. Numbers of (8..11)-digit terms are 1,4,24,129 (see b-file).

Zak Seidov, Table of n, a(n) for n = 1..158

A023087 Numbers n such that n and 3n are anagrams. A023093 Numbers n such that n and 9n are anagrams.

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}]

cp's OEIS Frontend

A023086 Numbers k such that k and 2*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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A023089 Numbers k such that k and 5*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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A336661 Numbers that have decimal expansion c(1)c(2)...c(n) with distinct digits that satisfy c(1) <> 0, c(1) is the largest digit, and for each i in 1..n there is j in 0..2 such that c(i) == 3*c(i-1) + j (mod 10) (with c(0): = c(n)).

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