A173770 -id:A173770 - cp's OEIS frontend

A309608 Digits of the 10-adic integer (-13/9)^(1/3).

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

Offset: 0

Seiichi Manyama, Aug 10 2019

			       7^3 == 3      (mod 10).
       7^3 == 43     (mod 10^2).
     307^3 == 443    (mod 10^3).
     307^3 == 4443   (mod 10^4).
   30307^3 == 44443  (mod 10^5).
  330307^3 == 444443 (mod 10^6).

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

Cf. A173770, A309600, A309613.

N=100; Vecrev(digits(lift(chinese(Mod((-13/9+O(2^N))^(1/3), 2^N), Mod((-13/9+O(5^N))^(1/3), 5^N)))), N)

def A309608(n)
  ary = [7]
  a = 7
  n.times{|i|
    b = (a + 3 * (9 * a ** 3 + 13)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309608(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + 3 * (9 * b(n-1)^3 + 13) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.

A093163 Primes of the form 40*R_k + 3, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

3, 43, 443, 444443, 444444443, 444444444443, 444444444444444444444444444443, 44444444444444444444444444444443

Offset: 1

Rick L. Shepherd, Mar 26 2004

nonn

Primes of the form (4*10^k - 13)/9; the next term a(9) is a 183-digit number. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 44...443.
Index entries for primes involving repunits

Cf. A056661 (corresponding k), A173770.

f[n_] := (4*10^n-13)/9; Select[f[Range[50]], PrimeQ] (* Amiram Eldar, Nov 23 2018 *)
Select[Table[FromDigits[PadLeft[{3},n,4]],{n,40}],PrimeQ] (* Harvey P. Dale, Dec 19 2022 *)

A104660 Number of distinct prime divisors of 44...443 (with n 4s).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 6, 3, 4, 3, 3, 6, 4, 1, 5, 1, 4, 3, 5, 3, 6, 4, 2, 6, 2, 2, 3, 5, 3, 4, 4, 4, 2, 2, 4, 4, 3, 4, 5, 6, 3, 3, 5, 2, 4, 3, 5, 4, 4, 3, 6, 4, 3, 6, 6, 4, 5, 4, 2, 4, 5, 2, 4, 5, 6, 4, 4, 3, 6, 5, 4, 5

Offset: 1

Parthasarathy Nambi, Apr 21 2005

			The number of distinct prime divisors of 43 is 1 (prime).
The number of distinct prime divisors of 443 is 1 (prime).
The number of distinct prime divisors of 4443 is 2.

Amiram Eldar, Table of n, a(n) for n = 1..199

Cf. A001221, A104518, A173770.

A104660 := proc(n) x := [3,seq(4,k=1..n)] ; add(op(i,x)*10^(i-1),i=1..nops(x)) ; numtheory[factorset](%) ; nops(%) ; end proc:
seq(A104660(n),n=1..45) ; # R. J. Mathar, Aug 23 2011

Table[PrimeNu[FromDigits[Join[Table[4, {n}], {3}]]], {n, 50}] (* Alonso del Arte, Aug 23 2011 *)
Table[PrimeNu[FromDigits[PadLeft[{3},n,4]]],{n,2,70}] (* Harvey P. Dale, Aug 22 2016 *)

a(n) = A001221(A173770(n+1)). - Amiram Eldar, Jan 25 2020

More terms from R. J. Mathar and Alonso del Arte, Aug 23 2011

More terms from Amiram Eldar, Jan 25 2020

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A309608 Digits of the 10-adic integer (-13/9)^(1/3).

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A093163 Primes of the form 40*R_k + 3, where R_k is the repunit (A002275) of length k.

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A104660 Number of distinct prime divisors of 44...443 (with n 4s).

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