A061242 Primes of the form 9*k - 1.

17, 53, 71, 89, 107, 179, 197, 233, 251, 269, 359, 431, 449, 467, 503, 521, 557, 593, 647, 683, 701, 719, 773, 809, 827, 863, 881, 953, 971, 1061, 1097, 1151, 1187, 1223, 1259, 1277, 1367, 1439, 1493, 1511, 1583, 1601, 1619, 1637, 1709, 1871, 1889, 1907

Offset: 1

Author

Amarnath Murthy, Apr 23 2001

Keywords

• nonn
• easy

Comments

Or, primes of the form 18k - 1. Corresponding values of k are in A138918. - Zak Seidov, Apr 03 2008

From Doug Bell, Mar 23 2009: (Start)

Conjecture: if a(n) = 9x - 1, the integer formed by the repeating digits in the decimal fraction x/a(n) is the smallest integer such that rotating the digits to the left produces a number which is (x+1)/x times larger.

Example: x = 2, a(n) = 17: 2/17 = 0.1176470588235294... repeating with a cycle of 16.

1176470588235294 * 3/2 = 1764705882352941, which is 1176470588235294 rotated to the left.

An additional conjecture is that the values of x from this sequence are the only values where rotating an integer one to the left produces a value (x+1)/x times as large. (End)

The last conjecture is false. For example, for x = 3 we have 230769*(4/3) = 307692, but 9*3-1 = 26 is not in the sequence. - Giovanni Resta, Jul 28 2015

Conjecture: Primes p such that ((x+1)^9-1)/x has 4 irreducible factors of degree 2 over GF(p). - Federico Provvedi, Jun 27 2018

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A061237, A061238, A061239, A061240, A061241 (p mod 9 = 1, 2, 4, 5 and 7), A138918 (18n - 1 is prime), A258663 (9n - 1 is prime).

Can be partitioned in disjoint subsequences A062343 (primes with sum of digits s = 8), A106758 (s = 17), A106764 (s = 26), A106770 (s = 35), A106776 (s = 44), A106782 (s = 53), A107617 (s = 62), etc.

Programs

Magma

[a: n in [0..250] | IsPrime(a) where a is 9*n - 1 ]; // Vincenzo Librandi, Jun 07 2015

Maple

select(isprime, [seq(18*i-1,i=1..1000)]); # Robert Israel, Sep 03 2014

Mathematica

Select[ Range[ 2500 ], PrimeQ[ # ] && Mod[ #, 9 ] == 8 & ]
Select[9*Range[300] - 1, PrimeQ]

PARI

select( {is(n)=n%9==8&&isprime(n)}, primes([1,2000])) \\ M. F. Hasler, Mar 10 2022

Python

from sympy import prime
A061242 = [p for p in (prime(n) for n in range(1,10**3)) if not (p+1) % 18]
# Chai Wah Wu, Sep 02 2014

Formula

A010888(a(n)) = 8. - Reinhard Zumkeller, Feb 25 2005

a(n) ~ 6n log n. - Charles R Greathouse IV, May 14 2025

Extensions

More terms from Robert G. Wilson v, May 10 2001

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Apr 30 2008

Edited by M. F. Hasler, Mar 10 2022