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.

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A063006 Coefficients in a 10-adic square root of 1.

Original entry on oeis.org

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

Views

Author

Robert Lozyniak (11(AT)onna.com), Aug 03 2001

Keywords

Comments

10-adic integer x=.....86760045215487480163574218751 satisfying x^3=x.
A "bug" in the decimal enumeration system: another square root of 1.
Let a,b be integers defined in A018247, A018248 satisfying a^2=a,b^2=b, obviously a^3=a,b^3=b; let c,d,e,f be integers defined in A091661, A063006, A091663, A091664 then c^3=c, d^3=d, e^3=e, f^3=f, c+d=1, a+e=1, b+f=1, b+c=a, d+f=e, a+f=c, a=f+1, b=e+1, cd=-1, af=-1, gh=-1 where -1=.....999999999. - Edoardo Gueglio (egueglio(AT)yahoo.it), Jan 28 2004
What about the 10-adic square roots of -1, -2, -3, 2, 3, 4, ...? They do not exist, unless the square really is a square (+1, +4, +9, +16, ...). Then there's a nontrivial square root; for example, sqrt(4)=...44002229693692923584436016426479909569025039672851562498. - Don Reble, Apr 25 2006

Examples

			...4218751^2 = ...0000001

References

  • K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973.

Links

Crossrefs

Another 10-adic root of 1 is given by A091661.
Cf. A075693.

Programs

  • Mathematica
    To calculate c, d, e, f use Mathematica algorithms for a, b and equations: c=a-b, d=1-c, e=b-1, f=a-1. - Edoardo Gueglio (egueglio(AT)yahoo.it), Jan 28 2004

Formula

(a_0 + a_1*10 + a_2*10^2 + a_3*10^3 + ... )^2 = 1 + 0*10 + 0*10^2 + 0*10^3 + ...
For n > 0, a(n) = 9 - A091661(n).

Extensions

More terms from Vladeta Jovovic, Aug 11 2001

A290372 10-adic integer x = ...5807 satisfying x^5 = x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 28 2017

Keywords

Comments

Also x^2 = A091661.

Examples

			     7^5 -    7 == 0 mod 10,
     7^5 -    7 == 0 mod 10^2,
   807^5 -  807 == 0 mod 10^3,
  5807^5 - 5807 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  2^(5^0) - 5^(2^0) ==    7 mod 10,
  2^(5^1) - 5^(2^1) ==    7 mod 10^2,
  2^(5^2) - 5^(2^2) ==  807 mod 10^3,
  2^(5^3) - 5^(2^3) == 5807 mod 10^4. (End)

Links

Crossrefs

Cf. A120817, A120818, A290373, A290374, A290375.
Cf. A091661, A018247.

Programs

  • Ruby
    def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s1
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290372(n)
  str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290372(100)

Formula

p = A120817 = ...186432, q = A018247 = ...890625, x = p - q = ...295807.

A290373 10-adic integer x = ...2943 satisfying x^5 = x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 28 2017

Keywords

Comments

Also x^2 = A091661.

Examples

			     3^5 -    3 == 0 mod 10,
    43^5 -   43 == 0 mod 10^2,
   943^5 -  943 == 0 mod 10^3,
  2943^5 - 2943 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  8^(5^0) - 5^(2^0) ==    3 mod 10,
  8^(5^1) - 5^(2^1) ==   43 mod 10^2,
  8^(5^2) - 5^(2^2) ==  943 mod 10^3,
  8^(5^3) - 5^(2^3) == 2943 mod 10^4. (End)

Links

Crossrefs

Cf. A120817, A120818, A290372, A290374, A290375.
Cf. A091661, A120818.

Programs

  • Ruby
    def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s2
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290373(n)
  str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290373(100)

Formula

p = A120818 = ...813568, q = A018247 = ...890625, x = p - q = ...922943.

A290374 10-adic integer x = ...7057 satisfying x^5 = x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 28 2017

Keywords

Comments

Also x^2 = A091661.

Examples

			     7^5 -    7 == 0 mod 10,
    57^5 -   57 == 0 mod 10^2,
    57^5 -   57 == 0 mod 10^3,
  7057^5 - 7057 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  2^(5^0) + 5^(2^0) ==    7 mod 10,
  2^(5^1) + 5^(2^1) ==   57 mod 10^2,
  2^(5^2) + 5^(2^2) ==   57 mod 10^3,
  2^(5^3) + 5^(2^3) == 7057 mod 10^4. (End)

Links

Crossrefs

x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), this sequence (...7057), A290375 (...4193).
Cf. A091661, A018247.

Programs

  • Ruby
    def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s1
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290374(n)
  str = (P(n) + Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290374(100)

Formula

p = A120817 = ...186432, q = A018247 = ...890625, x = p + q = ...077057.

A290375 10-adic integer x = ...4193 satisfying x^5 = x.

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 28 2017

Keywords

Comments

Also x^2 = A091661.

Examples

			     3^5 -    3 == 0 mod 10,
    93^5 -   93 == 0 mod 10^2,
   193^5 -  193 == 0 mod 10^3,
  4193^5 - 4193 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  8^(5^0) + 5^(2^0) ==    3 mod 10,
  8^(5^1) + 5^(2^1) ==   93 mod 10^2,
  8^(5^2) + 5^(2^2) ==  193 mod 10^3,
  8^(5^3) + 5^(2^3) == 4193 mod 10^4. (End)

Links

Crossrefs

x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), A290374 (...7057), this sequence (...4193).
Cf. A091661, A120818.

Programs

  • Ruby
    def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s2
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290375(n)
  str = (P(n) + Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290375(100)

Formula

p = A120818 = ...813568, q = A018247 = ...890625, x = p + q = ...704193.

A318135 The 10-adic integer a = ...1588948901 satisfying a^2 + 1 = b, b^2 + 1 = c, c^2 + 1 = d, d^2 + 1 = e, e^2 + 1 = f, and f^2 + 1 = a.

Original entry on oeis.org

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

Views

Author

Patrick A. Thomas, Aug 19 2018

Keywords

Comments

Data generated using MATLAB.
Conjecture: Let r(k) = the smallest positive residue of A003095(6*k+1) mod 10^(6*k+1). Then the first 2*k + 2 digits of r(k), reading from right to left, give the first 2*k + 2 digits of this 10-adic number. For example with k = 5, r(k) = 2121286728960294(201588948901) gives the first 12 digits correctly. - Peter Bala, Nov 14 2022

Examples

			901^2 + 1 == 802 (mod 10^3), 802^2 + 1 == 205 (mod 10^3), 205^2 + 1 == 26 (mod 10^3), 26^2 + 1 == 677 (mod 10^3), 677^2 + 1 == 330 (mod 10^3), and 330^2 + 1 == 901 (mod 10^3), so 1 0 9 comprise the sequence's first three terms.

Links

Crossrefs

Cf. A018247, A003095, A318136 (b), A318137 (c), A318138 (d), A318139 (e), A318140 (f).

A318136 The 10-adic integer b = ...9989107802 satisfying b^2 + 1 = c, c^2 + 1 = d, d^2 + 1 = e, e^2 + 1 = f, f^2 + 1 = a, and a^2 + 1 = b.

Original entry on oeis.org

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

Views

Author

Patrick A. Thomas, Aug 19 2018

Keywords

Comments

Data generated using MATLAB.

Examples

			802^2 + 1 = 205 (mod 10^3), 205^2 + 1 = 26 (mod 10^3), 26^2 + 1 = 677 (mod 10^3), 677^2 + 1 = 330 (mod 10^3), 330^2 + 1 = 901 (mod 10^3), and 901^2 + 1 = 802 (mod 10^3), so 2 0 8 comprise the sequence's first three terms.

Links

Crossrefs

Cf. A018247, A318135 (a), A318137 (c), A318138 (d), A318139 (e), A318140 (f).

A318137 The 10-adic integer c = ...9977271205 satisfying c^2 + 1 = d, d^2 + 1 = e, e^2 + 1 = f, f^2 + 1 = a, a^2 + 1 = b, and b^2 + 1 = c.

Original entry on oeis.org

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

Views

Author

Patrick A. Thomas, Aug 19 2018

Keywords

Comments

Data generated using MATLAB.

Examples

			205^2 + 1 == 26 (mod 10^3), 26^2 + 1 == 677 (mod 10^3), 677^2 + 1 == 330 (mod 10^3), 330^2 + 1 == 901 (mod 10^3), 901^2 + 1 == 802 (mod 10^3), and 802^2 + 1 == 205 (mod 10^3), so 5 0 2 comprise the sequence's first three terms.

Links

Crossrefs

Cf. A018247, A318135 (a), A318136 (b), A318138 (d), A318139 (e), A318140 (f).

A318138 The 10-adic integer d = ...8122152026 satisfying d^2 + 1 = e, e^2 + 1 = f, f^2 + 1 = a, a^2 + 1 = b, b^2 + 1 = c, and c^2 + 1 = d.

Original entry on oeis.org

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

Views

Author

Patrick A. Thomas, Aug 19 2018

Keywords

Comments

Data generated using MATLAB.

Examples

			26^2 + 1 == 677 (mod 10^3), 677^2 + 1 == 330 (mod 10^3), 330^2 + 1 == 901 (mod 10^3), 901^2 + 1 == 802 (mod 10^3), 802^2 + 1 == 205 (mod 10^3), and 205^2 + 1 == 26 (mod 10^3), so 6 2 0 comprise the sequence's first three terms.

Links

Crossrefs

Cf. A018247, A318135 (a), A318136 (b), A318137 (c), A318139 (e), A318140 (f)

A318139 The 10-adic integer e = ...3455904677 satisfying e^2 + 1 = f, f^2 + 1 = a, a^2 + 1 = b, b^2 + 1 = c, c^2 + 1 = d, and d^2 + 1 = e.

Original entry on oeis.org

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

Views

Author

Patrick A. Thomas, Aug 19 2018

Keywords

Comments

Data generated using MATLAB.

Examples

			677^2 + 1 == 330 (mod 10^3), 330^2 + 1 == 901 (mod 10^3), 901^2 + 1 = =802 (mod 10^3), 802^2 + 1 == 205 (mod 10^3), 205^2 + 1 == 26 (mod 10^3), and 26^2 + 1 == 677(mod10^3), so 7 7 6 comprise the sequence's first three terms.

Links

Crossrefs

Cf. A018247, A318135 (a), A318136 (b), A318137 (c), A318138 (d), A318140 (f).
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