This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181539 #37 Feb 10 2025 03:13:45
%S A181539 9,49,249,1249,18751,218751,781249,24218751,74218751,1425781249,
%T A181539 13574218751,163574218751,163574218751,19836425781249,19836425781249,
%U A181539 2480163574218751,12519836425781249,12519836425781249,487480163574218751,15487480163574218751,215487480163574218751,215487480163574218751
%N A181539 Smallest number m > 1 such that m^2 == 1 (mod 10^n).
%C A181539 a(n) > 10^floor(n/2).
%C A181539 All terms have last digit 1 or 9.
%C A181539 Squares of terms are listed in A085877.
%C A181539 Decimal representation of each term is formed by the reverse concatenation of initial terms of either A063006 or A091661.
%C A181539 Except for 3, there are no solutions for n>1 and m^2 == -1 (mod 10^n). See comment in A063006 under extensions. - _Robert G. Wilson v_, Jan 26 2013
%C A181539 If a(n)<(10^n)/2 then (10^n-a(n))^2 is also congruent (modulo 10^n), it is just not the least. - _Robert G. Wilson v_, Jan 26 2013
%H A181539 Max Alekseyev, <a href="/A181539/b181539.txt">Table of n, a(n) for n = 1..1000</a>
%F A181539 Let b(n) = A224474(n) (or equivalently b(n) = A224473(n)), then for n >= 3, there are eight solutions in [0,10^n) to x^2 == 1 (mod 10^n), namely x = 1, 5*10^(n-1) - 1, 5*10^(n-1) + 1, 10^n - 1, b(n), 10^n - b(n), |b(n) - 5*10^(n-1)|, and 10^n - |b(n) - 5*10^(n-1)|, so a(n) = min{b(n), |b(n) - 5*10^(n-1)|, 10^n - b(n)} < 25*10^(n-2). - _Jianing Song_, Sep 23 2024
%e A181539 1249^2 = 1560001 == 1 (mod 10^4), and there is no smaller m > 1 such that m^2 == 1 (mod 10^4). Hence a(4) = 1249.
%o A181539 (PARI) install(Zn_quad_roots, GGG);
%o A181539 a181539(n) = vecsort(Zn_quad_roots(10^n,0,-1)[2])[2]; \\ _Max Alekseyev_, Oct 13 2024
%Y A181539 Cf. A085610, A181607.
%Y A181539 Cf. A063006, A091661 (the two nontrivial 10-adic square roots of 1).
%Y A181539 Cf. A224473, A224474 (approximation of the two nontrivial 10-adic square roots of 1 up to powers of 10).
%K A181539 nonn
%O A181539 1,1
%A A181539 Kevin Batista (kevin762401(AT)yahoo.com), Oct 29 2010
%E A181539 a(2) through a(4), a(7) through a(11) corrected, comment added, example replaced by _Klaus Brockhaus_, Nov 01 2010
%E A181539 Edited by _N. J. A. Sloane_, Oct 29 2010, Nov 09 2010
%E A181539 Definition to avoid the constant sequence a(n)=1 constrained by _R. J. Mathar_, Nov 18 2010
%E A181539 a(1) corrected, terms a(13) onward added by _Max Alekseyev_, Dec 10 2012