OFFSET
1,1
COMMENTS
It is my conjecture that the integer formed by the repeating digits in the decimal fraction a(n)/(a(n)*9-1) is the smallest integer such that rotating the digits to the left produces a number which is ((a(n)+1)/a(n)) times larger.
Example: a(n) = 2: 2/17 = 0.1176470588235294... repeating with a cycle of 16.
1176470588235294 x (3/2) = 1764705882352941, which is 1176470588235294 rotated to the left.
An additional conjecture is that the values x in this sequence are the only values where rotating an integer one to the left produces a value (x+1)/x times as large. For example, the conjecture is that there are integers i that when rotated one to the left produce the value 3i/2, 7i/6 and 9i/8, but none that produce the value 2i/1, 4i/3, 5i/4, 6i/5 or 8i/7.
All of the terms in this sequence are even numbers that do not end with 4. (9n-1 is even for odd n and ends with 5 when the final digit of n = 4.) - Doug Bell, Jun 25 2015
The second conjecture is false. For example, 225806451612903*(8/7) = 258064516129032, or 45 * (6/5) = 54 or 230769*(4/3)=307692. - Giovanni Resta, Jul 28 2015
MATHEMATICA
Select[Range[2, 300], PrimeQ[9 # - 1] &] (* Vincenzo Librandi, Jun 07 2015 *)
PROG
(Magma) [n: n in [1..350] | IsPrime(9*n-1)]; // Vincenzo Librandi, Jun 07 2015
(PARI) is(n)=isprime(9*n-1) \\ Charles R Greathouse IV, Jun 06 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Doug Bell, Jun 07 2015
EXTENSIONS
More terms from Vincenzo Librandi, Jun 07 2015
STATUS
approved