

A007185


Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.
(Formerly M3940)


35



5, 25, 625, 625, 90625, 890625, 2890625, 12890625, 212890625, 8212890625, 18212890625, 918212890625, 9918212890625, 59918212890625, 259918212890625, 6259918212890625, 56259918212890625, 256259918212890625, 2256259918212890625, 92256259918212890625
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OFFSET

1,1


COMMENTS

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
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


REFERENCES

V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 2534.
Ya. I. Perelman, Algebra can be fun, pp. 9798.
C. P. Schut, Idempotents. Report AMR9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
C. P. Schut, Idempotents, Report AMR9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819823.
Index entries for sequences related to automorphic numbers


FORMULA

a(n) = 5^(2^n) mod 10^n.
a(n)^2 == a(n) (mod 10^n), that is, a(n) is an idempotent in Z[10^n].
a(n+1) = a(n)^2 mod 10^(n+1).  Eric M. Schmidt, Jul 28 2012
a(2n) = (3*a(n)^2  2*a(n)^3) mod 10^(2n).  Sylvie Gaudel, Mar 10 2018


EXAMPLE

625 is in the sequence because 625^2 = 390625, which ends in 625.
90625 is in the sequence because 90625^2 = 8212890625, which ends in 90625.
90635 is not in the sequence because 90635^2 = 8214703225, which does not end in 90635.


MAPLE

a:= n> 5&^(2^n) mod 10^n: seq(a(n), n=1..25); # Alois P. Heinz, Mar 11 2018


MATHEMATICA

Table[PowerMod[5, 2^n, 10^n], {n, 25}] (* Vincenzo Librandi, Jun 11 2016 *)


PROG

(Sage) [crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012
(PARI) A007185(n)=lift(Mod(5, 10^n)^2^n) \\ M. F. Hasler, Dec 05 2012
(MAGMA) [Modexp(5, 2^n, 10^n): n in [1..30]]; // Vincenzo Librandi, Jun 11 2016


CROSSREFS

A018247 gives the associated 10adic number.
A003226 = {0, 1} union (this sequence) union A016090.
Sequence in context: A082026 A101392 A078260 * A175852 A030995 A067270
Adjacent sequences: A007182 A007183 A007184 * A007186 A007187 A007188


KEYWORD

nonn,base


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David W. Wilson
Edited by David W. Wilson, Sep 26 2002
Further edited by N. J. A. Sloane, Jul 21 2010
Comment moved to name by Alonso del Arte, Mar 10 2018


STATUS

approved



