A007185 Automorphic numbers ending in digit 5: a(n) = 5^(2^n) mod 10^n.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base

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

Examples

			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.
		

References

• V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
• R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
• Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
• Ya. I. Perelman, Algebra can be fun, pp. 97-98.
• C. P. Schut, Idempotents. Report AM-R9101, 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 AM-R9101, 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. 819-823.
• Index entries for sequences related to automorphic numbers

Crossrefs

A018247 gives the associated 10-adic number.

A003226 = {0, 1} union (this sequence) union A016090.

Programs

Magma

[Modexp(5, 2^n, 10^n): n in [1..30]]; // Vincenzo Librandi, Jun 11 2016

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 *)

PARI

A007185(n)=lift(Mod(5,10^n)^2^n)  \\ M. F. Hasler, Dec 05 2012

Sage

[crt(1, 0, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012

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

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