cp's OEIS Frontend

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

%I A016090 #90 Feb 16 2025 08:32:33
%S A016090 6,76,376,9376,9376,109376,7109376,87109376,787109376,1787109376,
%T A016090 81787109376,81787109376,81787109376,40081787109376,740081787109376,
%U A016090 3740081787109376,43740081787109376,743740081787109376,7743740081787109376,7743740081787109376
%N A016090 a(n) = 16^(5^n) mod 10^n: Automorphic numbers ending in digit 6, with repetitions.
%C A016090 Also called congruent numbers.
%C A016090 a(n)^2 == a(n) (mod 10^n), that is, a(n) is idempotent of Z[10^n].
%C A016090 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
%C A016090 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
%D A016090 R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
%D A016090 V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
%D A016090 R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
%D A016090 Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4.
%D A016090 Ya. I. Perelman, Algebra can be fun, pp. 97-98.
%D A016090 A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.
%D A016090 C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
%H A016090 Eric M. Schmidt, <a href="/A016090/b016090.txt">Table of n, a(n) for n = 1..1000</a>
%H A016090 Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%H A016090 C. P. Schut, <a href="/A007185/a007185.pdf">Idempotents</a>, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
%H A016090 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AutomorphicNumber.html">Automorphic Number</a>
%H A016090 Xiaolong Ron Yu, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.10.No.10.pdf">Curious Numbers</a>, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
%H A016090 <a href="/index/Ar#automorphic">Index entries for sequences related to automorphic numbers</a>
%F A016090 a(n) = 16^(5^n) mod 10^n.
%F A016090 a(n+1) == 2*a(n) - a(n)^2 (mod 10^(n+1)). - _Eric M. Schmidt_, Jul 28 2012
%F A016090 a(n) = 6^(5^n) mod 10^n. - _Sylvie Gaudel_, Feb 17 2018
%F A016090 a(2*n) = (3*a(n)^2 - 2*a(n)^3) mod 10^(2*n). - _Sylvie Gaudel_, Mar 12 2018
%F A016090 a(n) = 6^5^(n-1) mod 10^n. - _M. F. Hasler_, Jan 26 2020
%F A016090 a(n) = 2^(10^n) mod 10^n for n >= 2. - _Peter Bala_, Nov 10 2022
%e A016090 a(5) = 09376 because 09376^2 == 87909376 ends in 09376.
%p A016090 [seq(16 &^ 5^n mod 10^n, n=1..22)]; # _Muniru A Asiru_, Mar 20 2018
%t A016090 Array[PowerMod[16, 5^#, 10^#] &, 18] (* _Michael De Vlieger_, Mar 13 2018 *)
%o A016090 (Sage) [crt(0, 1, 2^n, 5^n) for n in range(1, 1001)] # _Eric M. Schmidt_, Aug 18 2012
%o A016090 (PARI) A016090(n)=lift(Mod(6,10^n)^5^(n-1)) \\ _M. F. Hasler_, Dec 05 2012, edited Jan 26 2020
%o A016090 (Magma) [Modexp(16, 5^n, 10^n): n in [1..30]]; // _Bruno Berselli_, Mar 13 2018
%o A016090 (GAP) List([1..22], n->PowerModInt(16,5^n,10^n)); # _Muniru A Asiru_, Mar 20 2018
%Y A016090 A018248 gives the associated 10-adic number.
%Y A016090 A003226 = {0, 1} union A007185 union (this sequence).
%K A016090 nonn,base
%O A016090 1,1
%A A016090 _Robert G. Wilson v_, _David W. Wilson_
%E A016090 Edited by _David W. Wilson_, Sep 26 2002
%E A016090 Definition corrected by _M. F. Hasler_, Dec 05 2012