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%I #22 Jun 26 2017 05:53:38
%S 54,216,324,432,540,648,756,864,918,972,2232,4464,4554,5049,5454,5544,
%T 6696,7776,8928,9108,11070,19188,21951,22140,22221,29268,33210,38376,
%U 43443,43902,44280,44442,55350,56565,57564,58536,65853,66420,66663,73170,76752,77490,77787,80487,81180,86886,87804
%N Numbers n such that n * (x-1)/x produces a rotation of the digits in n for some value of x.
%C Numbers n where n * (x-1)/x produces a rotation that would have a first digit of zero are omitted.
%C Where n * (x-1)/x produces a rotation, x is a factor of n.
%C The first term where more than one value of x produces a rotation for a(n) * (x-1)/x is a(47) = 87804: 87804 * 8/9 = 78048 and 87804 * 11/12 = 80487. The first term where more than two values of x produce a rotation is a(186) = 857142: 857142 * 1/2 = 428571, 857142 * 2/3 = 571428, and 857142 * 5/6 = 714285.
%C The first term where a(n) * (x-1)/x produces a rotation that itself appears in this sequence is a(4) = 432: 432 * 3/4 = 324 = a(3).
%C If all of the digits in a(n) <= 4, then a(n)*2 also appears; if all of the digits in a(n) <= 3, then a(n)*3 also appears; if all of the digits in a(n) <= 2, then a(n)*4 also appears. Similarly, if each of the digits in a(n) are a multiple of some number k, then a(n)/k also appears.
%C Where ABC represents the digits in a(n), then ABCABC, ABCABCABC, ... also appear in the sequence with the same value(s) of x.
%H Doug Bell and Giovanni Resta, <a href="/A288626/b288626.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Doug Bell)
%H Doug Bell, <a href="/A288626/a288626_1.txt">Table of n, a(n), x/(x+1), a(n)*x/(x+1) for n = 1..1000</a>
%e a(1) = 54, 54 * 5/6 = 45;
%e a(9) = 918, 918 * 33/34 = 891.
%t ok[n_] := Block[{d = IntegerDigits[n], m, trg, t}, m = Length[d]; trg = FromDigits /@ Select[ RotateLeft[d, #] & /@ Range[m-1], First[#] > 0 &]; {} != Select[ trg, (t = n/#; Numerator[t]== 1 + Denominator[t]) &]]; Select[ Range[10^5], ok] (* _Giovanni Resta_, Jun 14 2017 *)
%Y Cf. A034089, A288669, A285936.
%K nonn,base
%O 1,1
%A _Doug Bell_, Jun 11 2017
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