%I #34 Aug 01 2019 18:27:42
%S 7,5,0,7,7,0,2,9,0,8,1,4,3,2,5,9,5,3,6,9,1,7,1,8,7,2,0,7,3,6,9,6,1,3,
%T 3,3,7,3,3,2,8,6,5,5,4,6,7,9,1,6,8,3,2,2,4,3,3,3,1,5,0,2,4,3,0,1,9,2,
%U 0,9,6,9,5,6,1,0,0,7,2,0,4,6,6,2,9,3,5,1
%N 10-adic integer x = ...7057 satisfying x^5 = x.
%C Also x^2 = A091661.
%H Seiichi Manyama, <a href="/A290374/b290374.txt">Table of n, a(n) for n = 0..9999</a>
%F p = A120817 = ...186432, q = A018247 = ...890625, x = p + q = ...077057.
%e 7^5 - 7 == 0 mod 10,
%e 57^5 - 57 == 0 mod 10^2,
%e 57^5 - 57 == 0 mod 10^3,
%e 7057^5 - 7057 == 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) == 57 mod 10^2,
%e 2^(5^2) + 5^(2^2) == 57 mod 10^3,
%e 2^(5^3) + 5^(2^3) == 7057 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 A290374(n)
%o str = (P(n) + Q(5, n)).to_s.reverse
%o (0..n).map{|i| str[i].to_i}
%o end
%o p A290374(100)
%Y x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), this sequence (...7057), A290375 (...4193).
%Y Cf. A091661, A018247.
%K nonn,base
%O 0,1
%A _Seiichi Manyama_, Jul 28 2017