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Numbers n such that n * x/(x-1) produces a rotation of the digits in n for some value of x.
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%I #20 Jun 26 2017 05:53:17

%S 45,162,243,324,405,486,567,648,729,891,2223,4446,4455,4545,4950,5445,

%T 6669,7767,8892,8910,10701,18819,19512,21402,22212,26829,32103,37638,

%U 39024,42804,43434,44424,53505,53658,56457,56556,58536,64206,66636,70731,74907,75276,77778,78048

%N Numbers n such that n * x/(x-1) produces a rotation of the digits in n for some value of x.

%C Numbers n where n * x/(x-1) produces a rotation that would have a first digit of zero are omitted.

%C Where n * x/(x-1) produces a rotation, (x-1) is a factor of n.

%C The first term where more than one value of x produces a rotation for a(n) * x/(x-1) is a(44) = 78048: 78048 * 9/8 = 87804 and 78048 * 33/32 = 80487.

%C The first term where a(n) * x/(x-1) produces a rotation that itself appears in this sequence is a(3) = 243: 243 * 4/3 = 324 = a(4).

%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="/A288669/b288669.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Doug Bell)

%H Doug Bell, <a href="/A288669/a288669.txt">Table of n, a(n), x/(x-1), a(n)*x/(x-1) for n = 1..1000</a>

%e a(1) = 45, 45 * 6/5 = 54;

%e a(11) = 2223, 2223 * 248/247 = 2232.

%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, A288626, A285936.

%K nonn,base

%O 1,1

%A _Doug Bell_, Jun 13 2017