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.
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
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
Edited by David W. Wilson, Sep 26 2002
Definition corrected by M. F. Hasler, Dec 05 2012
STATUS
approved