A016090 a(n) = 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions.

6, 76, 376, 9376, 9376, 109376, 7109376, 87109376, 787109376, 1787109376, 81787109376, 81787109376, 81787109376, 40081787109376, 740081787109376, 3740081787109376, 43740081787109376, 743740081787109376, 7743740081787109376, 7743740081787109376

Offset

1,1

Comments

Also called congruent numbers.

a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].

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 2^n and a(n) - 1 is divisible by 5^n. - Eric M. Schmidt, Aug 18 2012

References

R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.

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.

A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.

C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.

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

Formula

a(n) = 16^(5^n) mod 10^n.

a(n+1) == 2*a(n) - a(n)^2 (mod 10^(n+1)). - Eric M. Schmidt, Jul 28 2012

a(n) = 6^(5^n) mod 10^n. - Sylvie Gaudel, Feb 17 2018

a(2*n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2*n). - Sylvie Gaudel, Mar 12 2018

a(n) = 6^5^(n-1) mod 10^n. - M. F. Hasler, Jan 26 2020

a(n) = 2^(10^n) mod 10^n for n >= 2. - Peter Bala, Nov 10 2022

Example

a(5) = 09376 because 09376^2 == 87909376 ends in 09376.

Maple

[seq(16 &^ 5^n mod 10^n, n=1..22)]; # Muniru A Asiru, Mar 20 2018

Mathematica

Array[PowerMod[16, 5^#, 10^#] &, 18] (* Michael De Vlieger, Mar 13 2018 *)

Prog

(Sage) [crt(0, 1, 2^n, 5^n) for n in range(1, 1001)] # Eric M. Schmidt, Aug 18 2012
(PARI) A016090(n)=lift(Mod(6, 10^n)^5^(n-1)) \\ M. F. Hasler, Dec 05 2012, edited Jan 26 2020
(Magma) [Modexp(16, 5^n, 10^n): n in [1..30]]; // Bruno Berselli, Mar 13 2018
(GAP) List([1..22], n->PowerModInt(16, 5^n, 10^n)); # Muniru A Asiru, Mar 20 2018

Crossrefs

A018248 gives the associated 10-adic number.

A003226 = {0, 1} union A007185 union (this sequence).

Sequence in context: A350411 A263228 A229571 * A181343 A241072 A137132

Adjacent sequences: A016087 A016088 A016089 * A016091 A016092 A016093

Keyword

nonn,base

Author

Robert G. Wilson v, David W. Wilson

Extensions

Edited by David W. Wilson, Sep 26 2002

Definition corrected by M. F. Hasler, Dec 05 2012

Status

approved