A290373 - OEIS

%I #29 Aug 01 2019 18:27:35

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

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

%U 9,0,3,0,4,3,8,9,9,2,7,9,5,3,3,7,0,6,4,8

%N 10-adic integer x = ...2943 satisfying x^5 = x.

%C Also x^2 = A091661.

%H Seiichi Manyama, <a href="/A290373/b290373.txt">Table of n, a(n) for n = 0..9999</a>

%F p = A120818 = ...813568, q = A018247 = ...890625, x = p - q = ...922943.

%e      3^5 -    3 == 0 mod 10,

%e     43^5 -   43 == 0 mod 10^2,

%e    943^5 -  943 == 0 mod 10^3,

%e   2943^5 - 2943 == 0 mod 10^4.

%e From _Seiichi Manyama_, Aug 01 2019: (Start)

%e   8^(5^0) - 5^(2^0) ==    3 mod 10,

%e   8^(5^1) - 5^(2^1) ==   43 mod 10^2,

%e   8^(5^2) - 5^(2^2) ==  943 mod 10^3,

%e   8^(5^3) - 5^(2^3) == 2943 mod 10^4. (End)

%o (Ruby)

%o def P(n)

%o   s1, s2 = 2, 8

%o   n.times{|i|

%o     m = 10 ** (i + 1)

%o     (0..9).each{|j|

%o       k1, k2 = j * m + s1, (9 - j) * m + s2

%o       if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0

%o         s1, s2 = k1, k2

%o         break

%o       end

%o     }

%o   }

%o   s2

%o end

%o def Q(s, n)

%o   n.times{|i|

%o     m = 10 ** (i + 1)

%o     (0..9).each{|j|

%o       k = j * m + s

%o       if (k ** 2 - k) % (m * 10) == 0

%o         s = k

%o         break

%o       end

%o     }

%o   }

%o   s

%o end

%o def A290373(n)

%o   str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse

%o   (0..n).map{|i| str[i].to_i}

%o end

%o p A290373(100)

%Y Cf. A120817, A120818, A290372, A290374, A290375.

%Y Cf. A091661, A120818.

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Jul 28 2017