A225401 10-adic integer x such that x^3 = -7/9.

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

Offset: 0

Author

Aswini Vaidyanathan, May 07 2013

Keywords

• nonn
• base

Examples

			       3^3 == 7      (mod 10).
      53^3 == 77     (mod 10^2).
     753^3 == 777    (mod 10^3).
     753^3 == 7777   (mod 10^4).
   60753^3 == 77777  (mod 10^5).
  660753^3 == 777777 (mod 10^6).
		

Links

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

Crossrefs

Cf. A225410, A309600.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, 3*n,
      irem(b(n-1)+3*(9*b(n-1)^3+7), 10^n))
    end:
a:= n-> (b(n+1)-b(n))/10^n:
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2022

PARI

n=0;for(i=1,100,m=7*(10^i-1)/9;for(x=0,9,if(((n+(x*10^(i-1)))^3)%(10^i)==m,n=n+(x*10^(i-1));print1(x", ");break)))

PARI

N=100; Vecrev(digits(lift(chinese(Mod((-7/9+O(2^N))^(1/3), 2^N), Mod((-7/9+O(5^N))^(1/3), 5^N)))), N) \\ Seiichi Manyama, Aug 05 2019

Ruby

def A225401(n)
  ary = [3]
  a = 3
  n.times{|i|
    b = (a + 3 * (9 * a ** 3 + 7)) % (10 ** (i + 2))
    ary << (b - a) / (10 ** (i + 1))
    a = b
  }
  ary
end
p A225401(100) # Seiichi Manyama, Aug 13 2019

Formula

Define the sequence {b(n)} by the recurrence b(0) = 0 and b(1) = 3, b(n) = b(n-1) + 3 * (9 * b(n-1)^3 + 7) mod 10^n for n > 1, then a(n) = (b(n+1) - b(n))/10^n. - Seiichi Manyama, Aug 13 2019