A225458 -id:A225458 - cp's OEIS frontend

A309698 Digits of the 4-adic integer 3^(1/3).

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

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)): this sequence (k=4), A309699 (k=6), A309700 (k=8), A225458 (k=10).

Cf. A225404, A290563, A322931.

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

def A309698(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + a ** 3 - 3) % (4 ** (i + 2))
    ary << (b - a) / (4 ** (i + 1))
    a = b
  }
  ary
end
p A309698(100)

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

A309699 Digits of the 6-adic integer 5^(1/5).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Aug 13 2019

x   = ...513045,
x^2 = ...433521,
x^3 = ...051525,
x^4 = ...354241,
x^5 = ...000005.

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), this sequence (k=6), A309700 (k=8), A225458 (k=10).

Cf. A309448.

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

def A309699(n)
  ary = [5]
  a = 5
  n.times{|i|
    b = (a + a ** 5 - 5) % (6 ** (i + 2))
    ary << (b - a) / (6 ** (i + 1))
    a = b
  }
  ary
end
p A309699(100)

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

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

Original entry on oeis.org

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.

A193345 Digits occurring in A173616.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Jul 23 2011

a(n) = A173616(n) - 10*A173616(n-1).

This is the 10-adic integer x such that x^9 == (10^n-9) mod 10^n for all n. It is the 10's complement of A225458. - Aswini Vaidyanathan, May 11 2013

			1111^1111=.........8711; 111^111=........711;
10^(1-4)(8711-711)=8 ==> a(4)=8
Comment from _Aswini Vaidyanathan_, May 11 2013:
1^9 == 1 (mod 10).
11^9 == 91 (mod 100).
711^9 == 991 (mod 1000).
8711^9 == 9991 (mod 10000).
78711^9 == 99991 (mod 100000).
78711^9 == 999991 (mod 1000000).

Cf. A173616, A225458.

repunit[n_] := Sum[10^i, {i,0,n-1}]; a[n_] := 10^(1-n)(PowerMod[repunit[n], repunit[n],10^n] - PowerMod[repunit[n-1], repunit[n-1], 10^(n-1)]); Table[a[n],{n,200}]

n=0;for(i=1,100,m=(10^i-9);for(x=0,9,if(((n+(x*10^(i-1)))^9)%(10^i)==m,n=n+(x*10^(i-1));print1(x", ");break))) (From Aswini Vaidyanathan, May 11 2013)

Edited by N. J. A. Sloane, May 12 2013

cp's OEIS Frontend

A309698 Digits of the 4-adic integer 3^(1/3).

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A309699 Digits of the 6-adic integer 5^(1/5).

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

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A193345 Digits occurring in A173616.

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