%I #37 Aug 01 2019 18:27:28
%S 7,0,8,5,9,2,6,6,6,1,8,5,3,0,0,7,4,8,1,1,4,2,6,8,7,8,7,3,2,4,1,6,1,5,
%T 1,1,5,4,5,0,2,2,9,0,6,9,2,1,7,4,7,2,2,2,2,1,7,5,8,7,8,5,2,4,8,0,6,9,
%U 6,4,4,8,5,8,3,0,8,6,5,2,5,0,6,6,9,9,1,5
%N 10-adic integer x = ...5807 satisfying x^5 = x.
%C Also x^2 = A091661.
%H Seiichi Manyama, <a href="/A290372/b290372.txt">Table of n, a(n) for n = 0..9999</a>
%F p = A120817 = ...186432, q = A018247 = ...890625, x = p - q = ...295807.
%e 7^5 - 7 == 0 mod 10,
%e 7^5 - 7 == 0 mod 10^2,
%e 807^5 - 807 == 0 mod 10^3,
%e 5807^5 - 5807 == 0 mod 10^4.
%e From _Seiichi Manyama_, Aug 01 2019: (Start)
%e 2^(5^0) - 5^(2^0) == 7 mod 10,
%e 2^(5^1) - 5^(2^1) == 7 mod 10^2,
%e 2^(5^2) - 5^(2^2) == 807 mod 10^3,
%e 2^(5^3) - 5^(2^3) == 5807 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 s1
%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 A290372(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 A290372(100)
%Y Cf. A120817, A120818, A290373, A290374, A290375.
%Y Cf. A091661, A018247.
%K nonn,base
%O 0,1
%A _Seiichi Manyama_, Jul 28 2017