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

1,1

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.

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

Example

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

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 *)

Crossrefs

Cf. A034089, A288626, A285936.

Sequence in context: A347874 A234336 A173371 * A305069 A061658 A254147

Adjacent sequences: A288666 A288667 A288668 * A288670 A288671 A288672

Keyword

nonn,base

Author

Doug Bell, Jun 13 2017

Status

approved