A173766 -id:A173766 - cp's OEIS frontend

A093168 Primes of the form 30*R_k + 7, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

7, 37, 337, 333337, 3333333333333333333333333333333333333333333337

Offset: 1

Rick L. Shepherd, Mar 26 2004

nonn

Primes of the form (3*10^k + 33)/9. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 33...337.
Index entries for primes involving repunits

Cf. A056680 (corresponding k), A099411, A173766.

A104483 Number of distinct prime divisors of 33...337 (with n 3s).

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 2, 2, 2, 3, 3, 4, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 4, 5, 3, 4, 2, 4, 4, 5, 4, 4, 4, 8, 2, 2, 2, 3, 5, 4, 6, 6, 1, 5, 3, 4, 5, 4, 5, 8, 4, 2, 5, 5, 5, 2, 4, 4, 6, 4, 2, 4, 3, 4, 6, 3, 6, 9, 4, 7, 5, 3, 5, 7, 3, 6, 5, 6, 4, 7, 4, 5, 3, 5, 4

Offset: 0

Parthasarathy Nambi, Apr 18 2005

Number of distinct prime factors of (10^(n + 1) - 1)/3 + 4. - Stefan Steinerberger, Mar 06 2006

			The number of distinct prime divisors of 37 is 1 (prime).
The number of distinct prime divisors of 337 is 1 (prime).
The number of distinct prime divisors of 3337 is 2.

Amiram Eldar, Table of n, a(n) for n = 0..199

Cf. A001221, A173766.

Table[Length[FactorInteger[(10^(n + 1) - 1)/3 + 4]], {n, 1, 50}] (* Stefan Steinerberger, Mar 06 2006 *)

a(n) = A001221(A173766(n+1)). - Amiram Eldar, Jan 24 2020

More terms from Stefan Steinerberger, Mar 06 2006

More terms from Amiram Eldar, Jan 24 2020

A178769 a(n) = (5*10^n + 13)/9.

Original entry on oeis.org

2, 7, 57, 557, 5557, 55557, 555557, 5555557, 55555557, 555555557, 5555555557, 55555555557, 555555555557, 5555555555557, 55555555555557, 555555555555557, 5555555555555557, 55555555555555557, 555555555555555557, 5555555555555555557, 55555555555555555557, 555555555555555555557

Offset: 0

Bruno Berselli, Jun 13 2010

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Bruno Berselli, A property that includes the numbers of the form 5..57.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A165246 (..17, 117, 1117,..), A173193 (..27, 227, 2227,..), A173766 (..37, 337, 3337,..), A173772 (..47, 447, 4447,..), A067275 (..67, 667, 6667,..), A002281 (..77, 777, 7777,..), A173812 (..87, 887, 8887,..), A173833 (..97, 997, 9997,..).

Cf. A093143.

List([0..20], n -> (5*10^n+13)/9); # G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9: n in [0..20]]; // Vincenzo Librandi, Jun 06 2013

CoefficientList[Series[(2 - 15 x) / ((1 - x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{11,-10},{2,7},20] (* Harvey P. Dale, Feb 28 2017 *)

vector(20, n, n--; (5*10^n+13)/9) \\ G. C. Greubel, Jan 24 2019

[(5*10^n+13)/9 for n in (0..20)] # G. C. Greubel, Jan 24 2019

a(n)^(4*k+2) + 1 == 0 (mod 250) for n > 1, k >= 0.
G.f.: (2-15*x)/((1-x)*(1-10*x)).
a(n) - 11*a(n-1) + 10*a(n-2) = 0 (n > 1).
a(n) = a(n-1) + 5*10^(n-1) = 10*a(n-1) - 13 for n > 0.
a(n) = 1 + Sum_{i=0..n} A093143(i). - Bruno Berselli, Feb 16 2015
E.g.f.: exp(x)*(5*exp(9*x) + 13)/9. - Elmo R. Oliveira, Sep 09 2024

A309569 Digits of the 10-adic integer (11/3)^(1/3).

Original entry on oeis.org

3, 3, 2, 2, 5, 8, 6, 8, 3, 1, 7, 3, 2, 6, 1, 0, 0, 1, 3, 2, 5, 3, 3, 5, 5, 5, 0, 8, 8, 9, 0, 9, 1, 7, 3, 2, 9, 4, 3, 9, 9, 8, 3, 3, 0, 1, 4, 2, 7, 6, 9, 5, 1, 5, 9, 3, 2, 5, 3, 7, 3, 1, 4, 8, 9, 7, 0, 1, 9, 1, 1, 4, 1, 4, 7, 4, 9, 0, 7, 7, 1, 2, 4, 9, 3, 4, 3, 8, 0, 8, 1, 8, 9, 8, 3, 5, 5, 1, 9, 2

Offset: 0

Seiichi Manyama, Aug 10 2019

			       3^3 == 7      (mod 10).
      33^3 == 37     (mod 10^2).
     233^3 == 337    (mod 10^3).
    2233^3 == 3337   (mod 10^4).
   52233^3 == 33337  (mod 10^5).
  852233^3 == 333337 (mod 10^6).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A173766, A309600, A309641.

N=100; Vecrev(digits(lift(chinese(Mod((11/3+O(2^N))^(1/3), 2^N), Mod((11/3+O(5^N))^(1/3), 5^N)))), N)

def A309569(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + 9 * (3 * a ** 3 - 11)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309569(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + 9 * (3 * b(n-1)^3 - 11) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.

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A093168 Primes of the form 30*R_k + 7, where R_k is the repunit (A002275) of length k.

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A104483 Number of distinct prime divisors of 33...337 (with n 3s).

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A178769 a(n) = (5*10^n + 13)/9.

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A309569 Digits of the 10-adic integer (11/3)^(1/3).

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