%I A003226 M3752
%S A003226 0,1,5,6,25,76,376,625,9376,90625,109376,890625,2890625,7109376,12890625,
%T A003226 87109376,212890625,787109376,1787109376,8212890625,18212890625,
%U A003226 81787109376,918212890625,9918212890625,40081787109376,59918212890625
%N A003226 Automorphic numbers: n^2 ends with n.
%C A003226 Also called curious numbers.
%D A003226 V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
%D A003226 R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
%D A003226 Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
%D A003226 Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
%D A003226 Ya. I. Perelman, Algebra can be fun, pp. 97-98.
%D A003226 C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
%H A003226 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Link to a section of The World of Mathematics.</a>
%Y A003226 A003226 = {0,1} union A007105 union A001609.
%Y A003226 Cf. A035383, A052228, A033819.
%K A003226 nonn,base,nice,easy
%O A003226 1,2
%A A003226 njas
%E A003226 More terms from Michel ten Voorde (upquark@gmx.net), Apr 11 2001

%I A007185 M3940
%S A007185 5,25,625,625,90625,890625,2890625,12890625,212890625,8212890625,
%T A007185 18212890625,918212890625,9918212890625,59918212890625,259918212890625,
%U A007185 6259918212890625,56259918212890625,256259918212890625
%N A007185 Automorphic numbers ending in digit 5.
%C A007185 a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].
%D A007185 V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
%D A007185 R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
%D A007185 Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
%D A007185 Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
%D A007185 Ya. I. Perelman, Algebra can be fun, pp. 97-98.
%D A007185 C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
%F A007185 a(n) = 2^5^n mod 10^n.
%H A007185 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Link to a section of The World of Mathematics.</a>
%e A007185 a(5) = 90625 because 90625^2 == 8212890625 ends in 90625.
%Y A007185 A018247 gives associated 10-adic number.
%Y A007185 A003226 = {0,1} union A007105 union A001609.
%K A007185 nonn,base
%O A007185 1,1
%A A007185 njas
%E A007185 More terms from David W. Wilson (davidwwilson@attbi.com)

%I A016090
%S A016090 6,76,376,9376,9376,109376,7109376,87109376,787109376,1787109376,
%T A016090 81787109376,81787109376,81787109376,40081787109376,740081787109376,
%U A016090 3740081787109376,43740081787109376,743740081787109376
%N A016090 Automorphic numbers ending in digit 6.
%C A016090 a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].
%D A016090 V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
%D A016090 R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
%D A016090 Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
%D A016090 Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
%D A016090 Ya. I. Perelman, Algebra can be fun, pp. 97-98.
%D A016090 C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
%F A016090 a(n) = 16^5^n mod 10^n.
%H A016090 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Link to a section of The World of Mathematics.</a>
%e A016090 a(5) = 09376 because 09376^2 == 87909376 ends in 09376.
%Y A016090 A018248 gives associated 10-adic number.
%Y A016090 A003226 = {0,1} union A007105 union A001609.
%K A016090 nonn,base
%O A016090 1,1
%A A016090 Robert G. Wilson v (rgwv@kspaint.com), Dave Wilson (davidwwilson@attbi.com)

%I A018247
%S A018247 5,2,6,0,9,8,2,1,2,8,1,9,9,5,2,6,5,2,2,9,3,7,7,9,9,1,6,6,0,1,4,0,0,9,0,1,
%T A018247 6,9,8,0,3,2,3,2,4,3,2,4,7,5,5,0,0,0,1,1,8,3,6,8,0,8,5,9,0,5,6,6,1,2,6,0,
%U A018247 0,9,8,9,0,5,8,3,9,2,0,8,9,6,1,8,0,1,9,1,3,7,0,0,3,5,9,3,0,9,3,6,2,4,6,7
%N A018247 The 10-adic number x = ...8212890625 solves x^2 = x.
%D A018247 W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
%D A018247 M. Kraitchik, Sphinx, 1935, p. 1.
%H A018247 Anonymous, <a href="http://freespace.virgin.net/anthony.edey/automorph.htm">Automorphic numbers (2)</a>
%H A018247 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Automorphic numbers (1)</a>
%C A018247 10-adics a and b defined in A018247 and A018248 satisfy a^2=a, b^2=b, a+b=1, ab=0.
%Y A018247 A007185 gives associated automorphic numbers.
%Y A018247 Cf. A018248, A033819.
%K A018247 base,nonn
%O A018247 1,1
%A A018247 Yoshihide Tamori (yo@salk.edu).
%E A018247 More terms from David W. Wilson (davidwwilson@attbi.com). Comments from Michael Somos (somos@grail.cba.csuohio.edu).

%I A018248
%S A018248 6,7,3,9,0,1,7,8,7,1,8,0,0,4,7,3,4,7,7,0,6,2,2,0,0,8,3,3,9,8,5,9,9,0,9,8,
%T A018248 3,0,1,9,6,7,6,7,5,6,7,5,2,4,4,9,9,9,8,8,1,6,3,1,9,1,4,0,9,4,3,3,8,7,3,9,
%U A018248 9,0,1,0,9,4,1,6,0,7,9,1,0,3,8,1,9,8,0,8,6,2,9,9,6,4,0,6,9,0,6,3,7,5,3,2
%N A018248 The 10-adic number x = ...1787109376 solves x^2 = x.
%D A018248 W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
%D A018248 M. Kraitchik, Sphinx, 1935, p. 1.
%H A018248 Anonymous, <a href="http://freespace.virgin.net/anthony.edey/automorph.htm">Automorphic numbers (2)</a>
%H A018248 E. W. Weisstein, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Automorphic numbers (1)</a>
%C A018248 10-adics a and b defined in A018247 and A018248 satisfy a^2=a, b^2=b, a+b=1, ab=0.
%Y A018248 A016090 gives associated automorphic numbers.
%Y A018248 Cf. A018247, A033819.
%K A018248 base,nonn
%O A018248 1,1
%A A018248 Yoshihide Tamori (yo@salk.edu)
%E A018248 More terms from David W. Wilson (davidwwilson@attbi.com). Comments from Michael Somos (somos@grail.cba.csuohio.edu).

%I A033819
%S A033819 0,1,4,5,6,9,24,25,49,51,75,76,99,125,249,251,375,376,499,501,624,625,
%T A033819 749,751,875,999,1249,3751,4375,4999,5001,5625,6249,8751,9375,9376,
%U A033819 9999,18751,31249,40625,49999,50001,59375,68751,81249,90624,90625
%N A033819 Trimorphic numbers: n^3 ends with n.
%H A033819 E. W. Weisstein, <a href="http://mathworld.wolfram.com/TrimorphicNumber.html">Link to a section of The World of Mathematics.</a>
%e A033819 376^3 = 53157376 which ends with 376.
%t A033819 Do[x=Floor[N[Log[10, n],25]]+1; If[Mod[n^3, 10^x] == n,Print[n]], {n,1,10000}]
%Y A033819 Cf. A074194.
%K A033819 base,nonn
%O A033819 1,2
%A A033819 David W. Wilson (davidwwilson@attbi.com)
 
%I A074194
%S A074194 0,1,9004106619977392256259918212890624,9004106619977392256259918212890625,18008213239954784512519836425781249
%T A074194 31991786760045215487480163574218751,40995893380022607743740081787109375,49999999999999999999999999999999999,50000000000000000000000000000000001,59004106619977392256259918212890625
%U A074194 68008213239954784512519836425781249,81991786760045215487480163574218751,90995893380022607743740081787109375,90995893380022607743740081787109376,99999999999999999999999999999999999
%N A074194 Trimorphic numbers: a(n)^3 == a(n) (mod 10^35).
%C A074194 There 15 trimorphs mod 10^n for n >= 3.
%Y A074194 Cf. A033819.
%K A074194 fini,full,nonn,base,new
%O A074194 1,1
%A A074194 Zakir F. Seidov (seidovzf@yahoo.com), Sep 19 2002