A178769 -id:A178769 - cp's OEIS frontend

A104524 Number of distinct prime divisors of 55...557 (with n 5s).

2, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 1, 4, 3, 3, 3, 4, 2, 1, 2, 7, 4, 4, 4, 4, 5, 4, 4, 3, 2, 3, 5, 3, 4, 3, 7, 2, 4, 2, 4, 4, 8, 2, 4, 3, 6, 3, 2, 4, 7, 7, 7, 5, 5, 6, 7, 2, 4, 6, 3, 5, 5, 2, 5, 2, 4, 5, 5, 3, 9, 10, 6, 5, 6, 4, 4, 4, 5, 4, 5, 3, 6, 6, 4, 1

Offset: 1

Parthasarathy Nambi, Apr 20 2005

			The number of distinct prime divisors of 57 is 2.
The number of distinct prime divisors of 557 is 1 (prime).
The number of distinct prime divisors of 5557 is 1 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..192

Cf. A104484, A104483, A104518.

Cf. A001221, A178769.

A104524 := proc(n) local x ;x := [7,seq(5,k=1..n)] ; add(op(i,x)*10^(i-1),i=1..nops(x)) ; numtheory[factorset](%) ; nops(%) ; end proc: # R. J. Mathar, Aug 24 2011

Table[PrimeNu[(50*10^n + 13)/9], {n, 1, 50}](* G. C. Greubel, May 07 2017 *)
Table[PrimeNu[FromDigits[PadLeft[{7},n,5]]],{n,2,90}] (* Harvey P. Dale, Dec 03 2021 *)

a(n) = omega((5*10^(n+1)+13)/9); \\ Michel Marcus, May 08 2017

a(n) = A001221(A178769(n+1)). - R. J. Mathar, Aug 24 2011

More terms from Michel Marcus, May 08 2017

More terms from Amiram Eldar, Jan 25 2020

A309613 Digits of the 10-adic integer (13/9)^(1/3).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 10 2019

			       3^3 == 7      (mod 10).
      93^3 == 57     (mod 10^2).
     693^3 == 557    (mod 10^3).
    9693^3 == 5557   (mod 10^4).
   69693^3 == 55557  (mod 10^5).
  669693^3 == 555557 (mod 10^6).

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

Cf. A178769, A309600, A309608.

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 A309613(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + 3 * (9 * a ** 3 - 13)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309613(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, 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.

A173735 a(n) = (10^n + 26)/9.

Original entry on oeis.org

3, 4, 14, 114, 1114, 11114, 111114, 1111114, 11111114, 111111114, 1111111114, 11111111114, 111111111114, 1111111111114, 11111111111114, 111111111111114, 1111111111111114, 11111111111111114, 111111111111111114, 1111111111111111114, 11111111111111111114, 111111111111111111114

Offset: 0

Vincenzo Librandi, Feb 23 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..100 (term 0 added by Ivan Panchenko)
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002279, A178769.

[(10^n+26)/9: n in [0..20]]; // Ivan Panchenko, Nov 05 2013

CoefficientList[Series[(3-29*x)/((1-x)*(1-10*x)),{x,0,30}],x] (* Ivan Panchenko, Nov 05 2013 *)

A173735(n):=(10^n+26)/9$
makelist(A173735(n),n,0,20); /* Ivan Panchenko, Nov 03 2013 */

a(n) = a(n-1) + 10^(n-1) = 10*a(n-1) - 26, a(0)=3.
a(n) = 11*a(n-1) - 10*a(n-2). - Vincenzo Librandi, Jul 05 2012
a(n) = 2*(A002279(n-1) + 2). - Martin Ettl, Nov 12 2012
G.f.: (3-29*x)/((1-x)*(1-10*x)). - Ivan Panchenko, Nov 05 2013
From Elmo R. Oliveira, Jun 18 2025: (Start)
E.g.f.: exp(x)*(26 + exp(9*x))/9.
a(n) = 2*A178769(n-1) for n >= 1. (End)

a(0) from Ivan Panchenko, Nov 05 2013

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A104524 Number of distinct prime divisors of 55...557 (with n 5s).

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A309613 Digits of the 10-adic integer (13/9)^(1/3).

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A173735 a(n) = (10^n + 26)/9.

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