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A288626
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Numbers n such that n * (x-1)/x produces a rotation of the digits in n for some value of x.
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2
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54, 216, 324, 432, 540, 648, 756, 864, 918, 972, 2232, 4464, 4554, 5049, 5454, 5544, 6696, 7776, 8928, 9108, 11070, 19188, 21951, 22140, 22221, 29268, 33210, 38376, 43443, 43902, 44280, 44442, 55350, 56565, 57564, 58536, 65853, 66420, 66663, 73170, 76752, 77490, 77787, 80487, 81180, 86886, 87804
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OFFSET
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1,1
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COMMENTS
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Numbers n where n * (x-1)/x produces a rotation that would have a first digit of zero are omitted.
Where n * (x-1)/x produces a rotation, x is a factor of n.
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.
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).
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.
Where ABC represents the digits in a(n), then ABCABC, ABCABCABC, ... also appear in the sequence with the same value(s) of x.
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LINKS
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EXAMPLE
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a(1) = 54, 54 * 5/6 = 45;
a(9) = 918, 918 * 33/34 = 891.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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