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%I M3940 #74 Sep 08 2022 08:44:35
%S 5,25,625,625,90625,890625,2890625,12890625,212890625,8212890625,
%T 18212890625,918212890625,9918212890625,59918212890625,
%U 259918212890625,6259918212890625,56259918212890625,256259918212890625,2256259918212890625,92256259918212890625
%N Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.
%C Conjecture: For any m coprime to 10 and for any k, the density of n such that a(n) == k (mod m) is 1/m. - _Eric M. Schmidt_, Aug 01 2012
%C a(n) is the unique positive integer less than 10^n such that a(n) is divisible by 5^n and a(n) - 1 is divisible by 2^n. - _Eric M. Schmidt_, Aug 18 2012
%D V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
%D R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
%D Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
%D Ya. I. Perelman, Algebra can be fun, pp. 97-98.
%D C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Eric M. Schmidt, <a href="/A007185/b007185.txt">Table of n, a(n) for n = 1..1000</a>
%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%H C. P. Schut, <a href="/A007185/a007185.pdf">Idempotents</a>, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AutomorphicNumber.html">Automorphic Number</a>
%H Xiaolong Ron Yu, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.10.No.10.pdf">Curious Numbers</a>, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
%H <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%F a(n) = 5^(2^n) mod 10^n.
%F a(n)^2 == a(n) (mod 10^n), that is, a(n) is an idempotent in Z[10^n].
%F a(n+1) = a(n)^2 mod 10^(n+1). - _Eric M. Schmidt_, Jul 28 2012
%F a(2n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2n). - _Sylvie Gaudel_, Mar 10 2018
%e 625 is in the sequence because 625^2 = 390625, which ends in 625.
%e 90625 is in the sequence because 90625^2 = 8212890625, which ends in 90625.
%e 90635 is not in the sequence because 90635^2 = 8214703225, which does not end in 90635.
%p a:= n-> 5&^(2^n) mod 10^n: seq(a(n), n=1..25); # _Alois P. Heinz_, Mar 11 2018
%t Table[PowerMod[5, 2^n, 10^n], {n, 25}] (* _Vincenzo Librandi_, Jun 11 2016 *)
%o (Sage) [crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # _Eric M. Schmidt_, Aug 18 2012
%o (PARI) A007185(n)=lift(Mod(5,10^n)^2^n) \\ _M. F. Hasler_, Dec 05 2012
%o (Magma) [Modexp(5, 2^n, 10^n): n in [1..30]]; // _Vincenzo Librandi_, Jun 11 2016
%Y A018247 gives the associated 10-adic number.
%Y A003226 = {0, 1} union (this sequence) union A016090.
%K nonn,base
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_
%E Edited by _David W. Wilson_, Sep 26 2002
%E Further edited by _N. J. A. Sloane_, Jul 21 2010
%E Comment moved to name by _Alonso del Arte_, Mar 10 2018
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