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%I #31 Sep 08 2022 08:46:12
%S 2,6,8,10,12,20,22,26,28,30,40,48,50,52,56,58,62,66,72,76,78,80,86,90,
%T 92,96,98,106,108,118,122,128,132,136,140,142,152,160,166,168,176,178,
%U 180,182,190,208,210,212,220,222,230,232,238,246,252,260
%N Numbers n such that 9n-1 is prime.
%C 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.
%C Example: a(n) = 2: 2/17 = 0.1176470588235294... repeating with a cycle of 16.
%C 1176470588235294 x (3/2) = 1764705882352941, which is 1176470588235294 rotated to the left.
%C 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.
%C 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
%C 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
%F a(n) = A138918(n)*2.
%F a(n) = (A061242(n)+1)/9.
%t Select[Range[2, 300], PrimeQ[9 # - 1] &] (* _Vincenzo Librandi_, Jun 07 2015 *)
%o (Magma) [n: n in [1..350] | IsPrime(9*n-1)]; // _Vincenzo Librandi_, Jun 07 2015
%o (PARI) is(n)=isprime(9*n-1) \\ _Charles R Greathouse IV_, Jun 06 2017
%Y Cf. A138918, A061242.
%K nonn,easy
%O 1,1
%A _Doug Bell_, Jun 07 2015
%E More terms from _Vincenzo Librandi_, Jun 07 2015
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