A225445 -id:A225445 - cp's OEIS frontend

A309700 Digits of the 8-adic integer 7^(1/7).

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

Offset: 0

Seiichi Manyama, Aug 13 2019

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

Digits of the k-adic integer (k-1)^(1/(k-1)): A309698 (k=4), A309699 (k=6), this sequence (k=8), A225458 (k=10).

Cf. A225445.

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

def A309700(n)
  ary = [7]
  a = 7
  n.times{|i|
    b = (a + a ** 7 - 7) % (8 ** (i + 2))
    ary << (b - a) / (8 ** (i + 1))
    a = b
  }
  ary
end
p A309700(100)

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

A225448 10-adic integer x such that x^7 = -7.

Original entry on oeis.org

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

Offset: 0

Aswini Vaidyanathan, May 11 2013

This is the 10's complement of A225445.

			       7^7 == -7 (mod 10).
      57^7 == -7 (mod 10^2).
     657^7 == -7 (mod 10^3).
    7657^7 == -7 (mod 10^4).
   87657^7 == -7 (mod 10^5).
  987657^7 == -7 (mod 10^6).

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

Cf. A225445.

n=0;for(i=1,100,m=(10^i-7);for(x=0,9,if(((n+(x*10^(i-1)))^7)%(10^i)==m,n=n+(x*10^(i-1));print1(x", ");break)))

N=100; Vecrev(digits(lift(chinese(Mod((-7+O(2^N))^(1/7), 2^N), Mod((-7+O(5^N))^(1/7), 5^N)))), N) \\ Seiichi Manyama, Aug 06 2019

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A309700 Digits of the 8-adic integer 7^(1/7).

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A225448 10-adic integer x such that x^7 = -7.

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