A173768 -id:A173768 - cp's OEIS frontend

A093174 Primes of the form 40*R_k + 1, where R_k is the repunit (A002275) of length k.

41, 4441, 44444444441, 4444444444444444444444444441, 4444444444444444444444444444444444444444444444444444441, 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444441

Offset: 1

Rick L. Shepherd, Mar 27 2004

nonn

Primes of the form (4*10^k - 31)/9. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 44...441.
Index entries for primes involving repunits

Cf. A056681 (corresponding k), A173768.

Edited by N. J. A. Sloane, Feb 26 2010

A104659 Number of distinct prime divisors of 44...441 (with n 4s).

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 2, 4, 2, 1, 5, 3, 2, 6, 3, 3, 3, 3, 2, 4, 4, 4, 4, 4, 4, 5, 1, 2, 6, 4, 4, 6, 4, 4, 4, 5, 4, 8, 4, 4, 7, 3, 2, 7, 3, 7, 4, 6, 3, 4, 6, 2, 6, 1, 4, 7, 2, 5, 4, 4, 4, 6, 4, 2, 3, 6, 3, 5, 4, 3, 11, 5, 4, 4, 5, 7, 3, 4, 3, 5, 4, 4, 3, 3, 6, 8, 3, 4, 4, 2, 6, 6, 1, 7, 8, 4, 4, 7, 4, 6, 6, 4, 4, 5, 6

Offset: 1

Parthasarathy Nambi, Apr 21 2005, extended Aug 08 2010

There are very few primes in this sequence. 41 appears as the smallest prime divisor frequently. There are many semiprimes.
41 is prime.
4441 is prime.
44444 444441 is prime.
4444 444444 444444 444444 444441 is prime.
4444444444444444444444444444444444444444444444444444441 is prime.
Computed using www.alpertron.com.ar/ECM.HTM

			The number of distinct prime divisors of 441 is 2.
The number of distinct prime divisors of 44444444444444444444444444444441 is four.

Amiram Eldar, Table of n, a(n) for n = 1..199
Dario Alpern, Factorization using the Elliptic Curve Method
Makoto Kamada, Factorizations of 44...441.

Cf. A001221, A104564, A104517, A104484, A173768.

f[n_] := Length@ FactorInteger[(4*10^(n + 1) - 31)/9]; Array[f, 105] (* Robert G. Wilson v, Aug 09 2010 *)
PrimeNu/@Rest[FromDigits/@Table[PadLeft[{1},n,4],{n,110}]] (* Harvey P. Dale, Mar 16 2012 *)

a(n) = A001221(A173768(n+1)). - Amiram Eldar, Jan 24 2020

a(32) - a(105) from Robert G. Wilson v, Aug 09 2010

A309595 Digits of the 10-adic integer (-31/9)^(1/3).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 10 2019

			       1^3 == 1      (mod 10).
      81^3 == 41     (mod 10^2).
      81^3 == 441    (mod 10^3).
    1081^3 == 4441   (mod 10^4).
   11081^3 == 44441  (mod 10^5).
  811081^3 == 444441 (mod 10^6).

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

Cf. A173768, A309600, A309614.

N=100; Vecrev(digits(lift(chinese(Mod((-31/9+O(2^N))^(1/3), 2^N), Mod((-31/9+O(5^N))^(1/3), 5^N)))), N)

def A309595(n)
  ary = [1]
  a = 1
  n.times{|i|
    b = (a + 7 * (9 * a ** 3 + 31)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309595(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 1, b(n) = b(n-1) + 7 * (9 * b(n-1)^3 + 31) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n.

cp's OEIS Frontend

A093174 Primes of the form 40*R_k + 1, where R_k is the repunit (A002275) of length k.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A104659 Number of distinct prime divisors of 44...441 (with n 4s).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A309595 Digits of the 10-adic integer (-31/9)^(1/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Ruby

Formula