A258663 Numbers n such that 9n-1 is prime.

2, 6, 8, 10, 12, 20, 22, 26, 28, 30, 40, 48, 50, 52, 56, 58, 62, 66, 72, 76, 78, 80, 86, 90, 92, 96, 98, 106, 108, 118, 122, 128, 132, 136, 140, 142, 152, 160, 166, 168, 176, 178, 180, 182, 190, 208, 210, 212, 220, 222, 230, 232, 238, 246, 252, 260

Offset: 1

Author

Doug Bell, Jun 07 2015

Keywords

• nonn
• easy

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

Crossrefs

Cf. A138918, A061242.

Programs

Magma

[n: n in [1..350] |  IsPrime(9*n-1)]; // Vincenzo Librandi, Jun 07 2015

Mathematica

Select[Range[2, 300], PrimeQ[9 # - 1] &] (* Vincenzo Librandi, Jun 07 2015 *)

PARI

is(n)=isprime(9*n-1) \\ Charles R Greathouse IV, Jun 06 2017

Formula

a(n) = A138918(n)*2.

a(n) = (A061242(n)+1)/9.

Extensions

More terms from Vincenzo Librandi, Jun 07 2015