cp's OEIS Frontend

A290373 10-adic integer x = ...2943 satisfying x^5 = x.

%I A290373 #29 Aug 01 2019 18:27:35
%S A290373 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 A290373 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 A290373 9,0,3,0,4,3,8,9,9,2,7,9,5,3,3,7,0,6,4,8
%N A290373 10-adic integer x = ...2943 satisfying x^5 = x.
%C A290373 Also x^2 = A091661.
%H A290373 Seiichi Manyama, <a href="/A290373/b290373.txt">Table of n, a(n) for n = 0..9999</a>
%F A290373 p = A120818 = ...813568, q = A018247 = ...890625, x = p - q = ...922943.
%e A290373      3^5 -    3 == 0 mod 10,
%e A290373     43^5 -   43 == 0 mod 10^2,
%e A290373    943^5 -  943 == 0 mod 10^3,
%e A290373   2943^5 - 2943 == 0 mod 10^4.
%e A290373 From _Seiichi Manyama_, Aug 01 2019: (Start)
%e A290373   8^(5^0) - 5^(2^0) ==    3 mod 10,
%e A290373   8^(5^1) - 5^(2^1) ==   43 mod 10^2,
%e A290373   8^(5^2) - 5^(2^2) ==  943 mod 10^3,
%e A290373   8^(5^3) - 5^(2^3) == 2943 mod 10^4. (End)
%o A290373 (Ruby)
%o A290373 def P(n)
%o A290373   s1, s2 = 2, 8
%o A290373   n.times{|i|
%o A290373     m = 10 ** (i + 1)
%o A290373     (0..9).each{|j|
%o A290373       k1, k2 = j * m + s1, (9 - j) * m + s2
%o A290373       if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
%o A290373         s1, s2 = k1, k2
%o A290373         break
%o A290373       end
%o A290373     }
%o A290373   }
%o A290373   s2
%o A290373 end
%o A290373 def Q(s, n)
%o A290373   n.times{|i|
%o A290373     m = 10 ** (i + 1)
%o A290373     (0..9).each{|j|
%o A290373       k = j * m + s
%o A290373       if (k ** 2 - k) % (m * 10) == 0
%o A290373         s = k
%o A290373         break
%o A290373       end
%o A290373     }
%o A290373   }
%o A290373   s
%o A290373 end
%o A290373 def A290373(n)
%o A290373   str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
%o A290373   (0..n).map{|i| str[i].to_i}
%o A290373 end
%o A290373 p A290373(100)
%Y A290373 Cf. A120817, A120818, A290372, A290374, A290375.
%Y A290373 Cf. A091661, A120818.
%K A290373 nonn,base
%O A290373 0,1
%A A290373 _Seiichi Manyama_, Jul 28 2017