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A016090 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions. 31
6, 76, 376, 9376, 9376, 109376, 7109376, 87109376, 787109376, 1787109376, 81787109376, 81787109376, 81787109376, 40081787109376, 740081787109376, 3740081787109376, 43740081787109376, 743740081787109376 (list; graph; refs; listen; history; text; internal format)
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

Obviously some terms are listed with repetition (a(5)=a(4), a(11)=a(12)=a(13),...), so this sequence is not just the list of automorphic numbers. - M. F. Hasler, Dec 05 2012

REFERENCES

R. Cuculiere, Jeux Mathematiques, 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

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

EXAMPLE

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

PROG

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

(PARI) a(n) = lift(Mod(16, 10^n)^(5^n)) \\ M. F. Hasler, Dec 05 2012

CROSSREFS

A018248 gives the associated 10-adic number.

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

Sequence in context: A185289 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

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Last modified March 28 12:08 EDT 2017. Contains 284186 sequences.