A288669 Numbers n such that n * x/(x-1) produces a rotation of the digits in n for some value of x.

45, 162, 243, 324, 405, 486, 567, 648, 729, 891, 2223, 4446, 4455, 4545, 4950, 5445, 6669, 7767, 8892, 8910, 10701, 18819, 19512, 21402, 22212, 26829, 32103, 37638, 39024, 42804, 43434, 44424, 53505, 53658, 56457, 56556, 58536, 64206, 66636, 70731, 74907, 75276, 77778, 78048

Offset: 1

Author

Doug Bell, Jun 13 2017

Keywords

• nonn
• base

Comments

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

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

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.

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).

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.

Examples

			a(1) = 45, 45 * 6/5 = 54;
a(11) = 2223, 2223 * 248/247 = 2232.
		

Links

• Doug Bell and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Doug Bell)
• Doug Bell, Table of n, a(n), x/(x-1), a(n)*x/(x-1) for n = 1..1000

Crossrefs

Cf. A034089, A288626, A285936.

Programs

Mathematica

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 *)