Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #45 Feb 04 2019 14:26:51
%S 142857,285714,1402857,1428570,1428597,1429857,2857014,2857140,
%T 2859714,2985714,14002857,14028570,14028597,14029857,14285700,
%U 14285970,14285997,14298570,14298597,14299857,15623784,15843762,17438256,17562438,18243756,21584376,23784156,24375618,24381756
%N Numbers k such that k, 2*k, and 3*k are anagrams of each other.
%C 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.
%C From _Chai Wah Wu_, Feb 01 2019: (Start)
%C 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.
%C 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.
%C 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.
%C 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.
%C Not all terms with digits 0 and 9 are formed this way, see for instance the terms 137965842 and 157836042.
%C The first term where the first digit is 3 is a(1507) = 3051267489.
%C (End)
%C From _David A. Corneth_, Feb 02 2019: (Start)
%C Terms are multiples of 9.
%C 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)
%H Chai Wah Wu, <a href="/A323711/b323711.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%o (Java) char[] digits1, digits2, digits3;
%o int val1, val2, val3;
%o for (int value=10; value<25000000; value++) {
%o digits1 = Integer.toString(value).toCharArray();
%o digits2 = Integer.toString(2*value).toCharArray();
%o digits3 = Integer.toString(3*value).toCharArray();
%o if (digits1.length == digits3.length) {
%o Arrays.sort(digits1);
%o Arrays.sort(digits2);
%o Arrays.sort(digits3);
%o val1 = Integer.parseInt(new String(digits1));
%o val2 = Integer.parseInt(new String(digits2));
%o val3 = Integer.parseInt(new String(digits3));
%o if ((val1 == val2) && (val1 == val3)) {
%o System.out.print(value + ",");
%o }
%o }
%o }
%o (Python)
%o 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
%Y Subsequence of A023086, numbers where n and 2*n are anagrams.
%K easy,nonn,base
%O 1,1
%A _Darrah Chavey_, Jan 24 2019