A138918 -id:A138918 - cp's OEIS frontend

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

Amarnath Murthy, Apr 23 2001

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

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

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.

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

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

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

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

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

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

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

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

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

Original entry on oeis.org

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

Doug Bell, Jun 07 2015

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

Cf. A138918, A061242.

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

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

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

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

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

More terms from Vincenzo Librandi, Jun 07 2015

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Original entry on oeis.org

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A098876 Least k such that 3((6n)^k) - 1 is prime.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2523, 2, 2, 1, 1, 2, 1, 1, 1, 2, 3, 6, 63, 1, 50, 38, 2, 1, 1, 1, 79, 1, 1, 3, 1, 4, 1, 2, 2, 1, 6, 1, 1, 1, 5, 3, 1, 18, 1, 1, 11, 1, 1, 26, 3, 10, 1, 1, 4, 2, 2, 4, 1, 6, 1, 4, 54, 1, 10, 1, 3, 1, 2, 1, 1

Offset: 1

Pierre CAMI, Oct 13 2004

nonn

a(72) > 3830, and the sequence then continues: 6, 2, 7, 1, 27, 2, 3, 1, 7, 2, 1, 1, 4, 36, 346, 1, 1, 1, 1, 3, 6, 2, 1, 2, 444, ...

a(72) > 10^4. - Ray Chandler, Nov 13 2004

Cf. A098877, A138918.

f[n_] := Block[{k = 1}, While[ !PrimeQ[3*((6*n)^k) - 1], k++ ]; k]; Table[ f[n], {n, 71}] (* Robert G. Wilson v, Oct 21 2004 *)

a(A138918(n)) = 1. - Michel Marcus, Jul 28 2015

Corrected and extended by Robert G. Wilson v, Oct 22 2004

A246965 Numbers n such that 19*n-(n+19) is a prime.

Original entry on oeis.org

2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 21, 25, 26, 27, 29, 30, 32, 34, 37, 39, 40, 41, 44, 46, 47, 49, 50, 54, 55, 60, 62, 65, 67, 69, 71, 72, 77, 81, 84, 85, 89, 90, 91, 92, 96, 105, 106, 107, 111, 112, 116, 117, 120, 124, 127, 131, 132, 134, 135, 137, 145, 146

Offset: 1

Shanmuga Subramanian, Sep 08 2014

nonn

			17 = (19*2)-(19+2) is prime, so 2 is a term.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Select[Range[150],PrimeQ[18#-19]&] (* Harvey P. Dale, Jun 10 2016 *)

lista(nn) = {for (n=1, nn, if (isprime(18*n-19), print1(n, ", ")););} \\ Michel Marcus, Sep 09 2014

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[n for n in (2..200) if is_prime(18*n-19)] # Bruno Berselli, Sep 09 2014

a(n) = A138918(n)+1.

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A061242 Primes of the form 9*k - 1.

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

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A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

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A098876 Least k such that 3*((6*n)^k) - 1 is prime.

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A246965 Numbers n such that 19*n-(n+19) is a prime.

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A098876 Least k such that 3((6n)^k) - 1 is prime.