cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A104517 Number of distinct prime divisors of 55...1 (with n 5s).

Original entry on oeis.org

2, 2, 3, 2, 2, 2, 3, 2, 5, 4, 1, 1, 3, 2, 5, 3, 4, 2, 4, 5, 4, 5, 3, 2, 3, 3, 3, 5, 3, 4, 6, 4, 4, 2, 4, 4, 3, 3, 5, 2, 2, 3, 2, 3, 7, 4, 3, 2, 5, 4, 4, 4, 6, 4, 8, 5, 3, 4, 7, 3, 2, 3, 4, 4, 5, 5, 5, 5, 6, 3, 5, 4, 2, 4, 4, 6, 4, 3, 2, 2, 6, 3, 5, 7, 5, 3, 6, 3, 4, 6, 7, 7
Offset: 1

Views

Author

Parthasarathy Nambi, Apr 19 2005

Keywords

Comments

Number of distinct prime factors of (10^(n + 1) - 1)*5/9 - 4. - Stefan Steinerberger, Mar 06 2006

Examples

			The number of distinct prime divisors of 51 is 2 which is the first term in the sequence.
The number of distinct prime divisors of 551 is 2 which is the second term in the sequence.
The number of distinct prime divisors of 5551 is 3 which is the third term in the sequence.

Links

Crossrefs

Cf. A001221, A056684 (a(n)=1), A104484, A173804.

Programs

  • Magma
    [#PrimeDivisors((10^(n+1)-1)*5 div 9-4): n in [1..80]]; // Vincenzo Librandi, Mar 09 2018
  • Maple
    f:= n -> nops(numtheory:-factorset( (10^(n + 1) - 1)*5/9 - 4)):
map(f, [$1..92]); # Robert Israel, Mar 08 2018
  • Mathematica
    Table[Length[FactorInteger[(10^(n + 1) - 1)*5/9 - 4]], {n, 1, 50}] (* Stefan Steinerberger, Mar 06 2006 *)
  • PARI
    a(n) = omega((10^(n + 1) - 1)*5/9 - 4); \\ Michel Marcus, Mar 09 2018

Formula

a(n) = A001221(A173804(n+1)). - Amiram Eldar, Jan 24 2020

Extensions

More terms from Stefan Steinerberger, Mar 06 2006
a(51)-a(92), and offset corrected, by Robert Israel, Mar 08 2018

A309611 Digits of the 10-adic integer (-41/9)^(1/3).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 10 2019

Keywords

Examples

			       1^3 == 1      (mod 10).
      51^3 == 51     (mod 10^2).
     351^3 == 551    (mod 10^3).
    4351^3 == 5551   (mod 10^4).
    4351^3 == 55551  (mod 10^5).
  304351^3 == 555551 (mod 10^6).

Links

Crossrefs

Cf. A173804, A309600, A309610.

Programs

  • PARI
    N=100; Vecrev(digits(lift(chinese(Mod((-41/9+O(2^N))^(1/3), 2^N), Mod((-41/9+O(5^N))^(1/3), 5^N)))), N)
  • Ruby
    def A309611(n)
  ary = [1]
  a = 1
  n.times{|i|
    b = (a + 7 * (9 * a ** 3 + 41)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A309611(100)

Formula

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 + 41) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n

A322927 Expansion of x*(1 + 5*x + 40*x^2)/((1 - x^2)*(1 - 10*x^2)).

Original entry on oeis.org

0, 1, 5, 51, 55, 551, 555, 5551, 5555, 55551, 55555, 555551, 555555, 5555551, 5555555, 55555551, 55555555, 555555551, 555555555, 5555555551, 5555555555, 55555555551, 55555555555, 555555555551, 555555555555, 5555555555551, 5555555555555, 55555555555551
Offset: 0

Views

Author

Vincenzo Librandi, Mar 17 2019

Keywords

Links

Crossrefs

Bisections give: A002279 (even part), A173804 (odd part).

Programs

  • Magma
    I:=[0, 1, 5, 51]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]];
  • Maple
    seq(coeff(series(x*(1+5*x+40*x^2)/((1-x^2)*(1-10*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Mar 17 2019
  • Mathematica
    CoefficientList[Series[x (1 + 5 x + 40 x^2) / (10 x^4 - 11 x^2 + 1), {x, 0, 25}], x]

Formula

G.f.: x*(1 + 5*x + 40*x^2)/((1 - x^2)*(1 - 10*x^2)).
a(n) = 11*a(n-2) - 10*a(n-4).
a(n) = 5*(10^n - 1)/9 for n even; a(n) = (5*10^n - 41)/9 otherwise.
Showing 1-3 of 3 results.

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