A023086 -id:A023086 - cp's OEIS frontend

A159816 Seven-digit terms in A023086.

1025874, 1028574, 1042587, 1042857, 1052874, 1054287, 1072854, 1074285, 1078524, 1078542, 1085274, 1085427, 1087254, 1087425, 1087524, 1087542, 1207854, 1208754, 1240785, 1240875, 1245789, 1245879, 1247589, 1247859, 1248579, 1248759, 1250874, 1254087, 1257489

Offset: 1

Zak Seidov, Apr 22 2009

All 288 terms have only two sets of digits: {{0,1,2,4,5,7,8},{1,2,4,5,7,8,9}} with exactly equal numbers of both sets = 144.
There are six 7-d numbers n such that n, 2*n and 4*n are anagrams, that is intersection of 7-d subsequences in A023086 and A023088: 1294857, 1428507, 1428570, 1428705, 1429857, 1492857.
These are all "norep" numbers, i.e., numbers with any repetitive digit are not permitted. - Harvey P. Dale, Oct 30 2011

			a(1)=1025874 because 1025874 and 2*1025874=2051748 both use the same set of digits {0,1,2,4,5,7,8};
a(21)=1245789 because 1245789 and 2*1245789=2491578 both use the same set of digits {1,2,4,5,7,8,9}.

Zak Seidov, Table of n, a(n) for n = 1..288

Cf. A023086 (k and 2*k are anagrams), A023088 (k and 4*k are anagrams).

anaQ[n_]:=Max[DigitCount[n]]==1&&Union[IntegerDigits[n]] == Union[ IntegerDigits[2n]]; Select[Range[1000000,1250000],anaQ] (* Harvey P. Dale, Oct 30 2011 *)

A159827 a(n) = number of n-digit terms in A023086.

Original entry on oeis.org

1, 0, 0, 0, 0, 12, 288, 4032, 47904, 398736, 2668464

Offset: 1

Zak Seidov, Apr 22 2009

1. There are no terms in A023086 with number of digits in the range 2..5: a(2..5)=0, then a(6)=12, a(7)=288 etc.

2. Minimal and maximal n-digit terms (n>=6) in A023086 are (10^(n-1)+25874) and (5*10^(n-1)-71429).

A023086 n and 2*n are anagrams. A159816 Seven-digit terms in A023086.

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 *)

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 *)

A323711 Numbers k such that k, 2k, and 3k are anagrams of each other.

Original entry on oeis.org

142857, 285714, 1402857, 1428570, 1428597, 1429857, 2857014, 2857140, 2859714, 2985714, 14002857, 14028570, 14028597, 14029857, 14285700, 14285970, 14285997, 14298570, 14298597, 14299857, 15623784, 15843762, 17438256, 17562438, 18243756, 21584376, 23784156, 24375618, 24381756

Offset: 1

Darrah Chavey, Jan 24 2019

We assume entries have no leading zeros, so that n = 53617824 is not in the sequence, even though 2*n = 107235648 and 3*n = 160853472 are anagrams of 053617824.
From Chai Wah Wu, Feb 01 2019: (Start)
The first digit of terms is either 1, 2 or 3. Numbers of the form 140..028570..0 and 29..98570..0140..0 are terms where the number of 9's and 0's can be zero.
More generally, let a number n be written in decimal as xxxzzz where x and z are arbitrary digits and xxx, zzz are not empty strings. Let m be the number that is written as zzz in decimal and k be the least power of 10 that is strictly greater than m.
If 3*m < k, then n is a term if and only if xxx0..0zzz0..0 is a term. Note that this condition is satisfied if the first digit of m is 0, 1 or 2.
If 2*k <= 3*m, then n is a term if and only if xxx9..9zzz0..0 is a term. Note that this condition is satisfied if the first digit of m is 7, 8, or 9.
Not all terms with digits 0 and 9 are formed this way, see for instance the terms 137965842 and 157836042.
The first term where the first digit is 3 is a(1507) = 3051267489.
(End)
From David A. Corneth, Feb 02 2019: (Start)
Terms are multiples of 9.
Proof: as 3*k and k have the same digits, k is divisible by 3. If k isn't divisible by 9 then it has a different digital sum from 3*k. Therefore, k is divisible by 9. (End)

			The first entry, 142857, is well known for having n, 2*n, 3*n, 4*n, 5*n and 6*n all being anagrams. The next two numbers for which that happens are 1428570 and 1429857.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subsequence of A023086, numbers where n and 2*n are anagrams.

char[] digits1, digits2, digits3;
int val1, val2, val3;
for (int value=10; value<25000000; value++) {
     digits1 = Integer.toString(value).toCharArray();
     digits2 = Integer.toString(2*value).toCharArray();
     digits3 = Integer.toString(3*value).toCharArray();
     if (digits1.length == digits3.length) {
          Arrays.sort(digits1);
          Arrays.sort(digits2);
          Arrays.sort(digits3);
          val1 = Integer.parseInt(new String(digits1));
          val2 = Integer.parseInt(new String(digits2));
          val3 = Integer.parseInt(new String(digits3));
          if ((val1 == val2) && (val1 == val3)) {
               System.out.print(value + ",");
          }
     }
}

A323711_list = [n for n in range(9,10**7,9) if sorted(str(n)) == sorted(str(2*n)) == sorted(str(3*n))] # Chai Wah Wu, Feb 02 2019

cp's OEIS Frontend

A159816 Seven-digit terms in A023086.

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A159827 a(n) = number of n-digit terms in A023086.

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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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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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A323711 Numbers k such that k, 2k, and 3k are anagrams of each other.