A309724 Digits of the 8-adic integer (1/7)^(1/7).

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

Offset: 0

Author

Seiichi Manyama, Aug 14 2019

Keywords

• nonn
• base

Links

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

Crossrefs

Digits of the k-adic integer (1/(k-1))^(1/(k-1)): A309722 (k=4), A309723 (k=6), this sequence (k=8), A225464 (k=10).

Cf. A309700.

Programs

PARI

N=100; Vecrev(digits(lift((1/7+O(2^(3*N)))^(1/7)), 8), N)

Ruby

def A309724(n)
  ary = [7]
  a = 7
  n.times{|i|
    b = (a + 7 * (7 * a ** 7 - 1)) % (8 ** (i + 2))
    ary << (b - a) / (8 ** (i + 1))
    a = b
  }
  ary
end
p A309724(100)

Formula

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 7, b(n) = b(n-1) + 7 * (7 * b(n-1)^7 - 1) mod 8^n for n > 1, then a(n) = (b(n+1) - b(n))/8^n.