A258663 -id:A258663 - 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

A138918 Numbers n such that 18n-1 is prime.

Original entry on oeis.org

1, 3, 4, 5, 6, 10, 11, 13, 14, 15, 20, 24, 25, 26, 28, 29, 31, 33, 36, 38, 39, 40, 43, 45, 46, 48, 49, 53, 54, 59, 61, 64, 66, 68, 70, 71, 76, 80, 83, 84, 88, 89, 90, 91, 95, 104, 105, 106, 110, 111, 115, 116, 119, 123, 126, 130, 131, 133, 134, 136, 144, 145, 148, 150

Offset: 1

Zak Seidov, Apr 03 2008

Corresponding primes are in A061242.

No terms in this sequence end with 2 or 7 (18n-1 ends with 5 when the last digit of n is 2 or 7). - David Garber, Jun 25 2015

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A061242, A258663.

[n: n in [0..150] | IsPrime(18*n-1)]; // Vincenzo Librandi, Jun 27 2015

Select[Range[200],PrimeQ[18#-1]&]  (* Harvey P. Dale, Mar 09 2011 *)

for(n=1,10^3,if(isprime(18*n-1),print1(n,", "))) \\ Derek Orr, Sep 03 2014

from gmpy2 import divexact, t_mod
from sympy import prime
A138918 = [divexact(p+1,18) for p in (prime(n) for n in range(1,10**6)) if not t_mod(p+1,18)] # Chai Wah Wu, Sep 02 2014

A337922 Numbers k such that when the first digit of k is shifted to the end the result is 3*k/2.

Original entry on oeis.org

1176470588235294, 2352941176470588, 3529411764705882, 4705882352941176, 5882352941176470, 11764705882352941176470588235294, 23529411764705882352941176470588, 35294117647058823529411764705882, 47058823529411764705882352941176, 58823529411764705882352941176470

Offset: 1

Amiram Eldar, Jan 29 2021

The problem of finding the least number in this sequence was suggested by the Polish-British mathematician and historian Jacob Bronowski (1908-1974).

Anderson (1988) credited the problem to the British mathematician John Edensor Littlewood (1885-1977). The solution to the problem was given in 1955 by the British mathematician Dudley Ernest Littlewood (1903-1979), a student of J. E. Littlewood (but they were not related).

			1176470588235294 is a term since 1764705882352941 = 3*1176470588235294/2.

Jacob Bronowski, New Statesman and Nation, Vol. 39, Dec. 24, 1949, p. 761.
Dan Pedoe, The Gentle Art of Mathematics, Macmillan, 1960, p. 11.

Amiram Eldar, Table of n, a(n) for n = 1..310
Oliver D. Anderson, On Littlewood's Little Puzzle, Teaching Mathematics and its Applications: An International Journal of the IMA, Vol. 7, No. 3 (1988), pp. 144-146.
J. H. Clarke, Note 2298. A Digital Puzzle, The Mathematical Gazette, Vol. 36, No. 318 (1952), p. 276.
Keith Devlin, Micro-Maths, Mathematical problems and theorems to consider and solve on a computer, Macmillan, 1984, pp. 38-39.
D. E. Littlewood, Note 2494. On Note 2298: a digital puzzle, The Mathematical Gazette, Vol. 39, No. 327 (1955), p. 58.
Joseph S. Madachy, Recreational Mathematics, The Fibonacci Quarterly, Vo. 6, No. 6 (1968), pp. 385-398. See page 389.
Math Stackexchange, Finding the Dr. Bronowski's number, 2015.
Sidney Penner, Problem E 1530, Elementary Problems and Solutions, The American Mathematical Monthly, Vol. 69, No. 7 (1962), p. 667; J. W. Ellis and others, Placing the First Digit Last, Solution to Problem E 1530, ibid., Vol. 70, No. 4 (1963), pp. 441-442.
D. G. Rogers, Jacob Bronowski (1908-1974) The Mathematical Gazette and retrodigitisation, The Mathematical Gazette, Vol. 92, No. 525 (2008), pp. 476-479; alternative link.
Wikipedia, Transposable integer.

Cf. A061242, A166320, A258663.

concat[n_, m_] := NestList[FromDigits[Join[{#}, IntegerDigits[n]]] &, n, m]; s = Range[2, 10, 2]*(10^16 - 1)/17; Union @ Flatten[concat[#, 2] & /@ s]

The decimal digits of the first 5 terms are the periodic parts of the decimal expansions of 2/17, 4/17, 6/17, 8/17 and 10/17. The next terms are all the concatenations of each of these terms with itself an integral number of times (Anderson, 1988).

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A061242 Primes of the form 9*k - 1.

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A138918 Numbers n such that 18n-1 is prime.

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A337922 Numbers k such that when the first digit of k is shifted to the end the result is 3*k/2.

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