A309724 -id:A309724 - cp's OEIS frontend

A309722 Digits of the 4-adic integer (1/3)^(1/3).

3, 0, 3, 2, 1, 1, 0, 1, 2, 2, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 2, 3, 2, 2, 3, 2, 3, 3, 1, 1, 2, 0, 1, 3, 0, 0, 2, 3, 2, 2, 2, 0, 0, 0, 0, 0, 3, 2, 0, 2, 0, 2, 0, 0, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 0, 2, 3, 1, 0, 0, 3, 3, 2, 3, 3, 3, 0, 3, 1, 3, 2, 3, 2, 2, 1, 2, 0, 3, 2, 0, 2, 3, 0, 0, 2, 0, 3, 3, 0

Offset: 0

Seiichi Manyama, Aug 14 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Hensel's Lemma.

Digits of the k-adic integer (1/(k-1))^(1/(k-1)): this sequence (k=4), A309723 (k=6), A309724 (k=8), A225464 (k=10).

Cf. A225411, A309698.

N=100; Vecrev(digits(lift((1/3+O(2^(2*N)))^(1/3)), 4), N)

def A309722(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + 3 * (3 * a ** 3 - 1)) % (4 ** (i + 2))
    ary << (b - a) / (4 ** (i + 1))
    a = b
  }
  ary
end
p A309722(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + 3 * (3 * b(n-1)^3 - 1) mod 4^n for n > 1, then a(n) = (b(n+1) - b(n))/4^n.

A309723 Digits of the 6-adic integer (1/5)^(1/5).

Original entry on oeis.org

5, 0, 4, 5, 5, 1, 5, 4, 1, 2, 0, 3, 5, 3, 2, 5, 0, 0, 5, 4, 5, 3, 0, 3, 4, 2, 0, 0, 1, 4, 0, 0, 1, 2, 2, 3, 3, 4, 5, 4, 4, 5, 3, 1, 1, 5, 2, 0, 5, 4, 2, 2, 5, 4, 0, 2, 5, 5, 4, 2, 1, 1, 4, 2, 2, 4, 5, 4, 1, 5, 0, 1, 0, 4, 2, 1, 3, 4, 0, 1, 2, 0, 0, 3, 3, 1, 3, 4, 2, 3, 4, 2, 5, 0, 3, 3, 0, 1, 2, 0, 4

Offset: 0

Seiichi Manyama, Aug 14 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Hensel's Lemma.

Digits of the k-adic integer (1/(k-1))^(1/(k-1)): A309722 (k=4), this sequence (k=6), A309724 (k=8), A225464 (k=10).

Cf. A309699.

N=100; Vecrev(digits(lift(chinese(Mod((1/5+O(2^N))^(1/5), 2^N), Mod((1/5+O(3^N))^(1/5), 3^N))), 6), N)

def A309723(n)
  ary = [5]
  a = 5
  n.times{|i|
    b = (a + 5 * (5 * a ** 5 - 1)) % (6 ** (i + 2))
    ary << (b - a) / (6 ** (i + 1))
    a = b
  }
  ary
end
p A309723(100)

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 5, b(n) = b(n-1) + 5 * (5 * b(n-1)^5 - 1) mod 6^n for n > 1, then a(n) = (b(n+1) - b(n))/6^n.

cp's OEIS Frontend

A309722 Digits of the 4-adic integer (1/3)^(1/3).

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