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A030466 Squares that are concatenations of two consecutive nonzero numbers.

%I A030466 #67 Dec 28 2021 14:18:44
%S A030466 183184,328329,528529,715716,60996100,1322413225,4049540496,
%T A030466 106755106756,453288453289,20661152066116,29752082975209,
%U A030466 2214532822145329,2802768328027684,110213248110213249,110667555110667556,147928995147928996,178838403178838404,226123528226123529
%N A030466 Squares that are concatenations of two consecutive nonzero numbers.
%D A030466 British Mathematical Olympiad, 1993, Round 1, Question 1: "Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits of n. (Thus n might look like 123124, although this is not a square.)"
%D A030466 Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of the British Mathematical Olympiad 1993, page 164.
%H A030466 Iain Fox, <a href="/A030466/b030466.txt">Table of n, a(n) for n = 1..10471</a>.
%H A030466 British Mathematical Olympiad 1993, Round 1, <a href="https://bmos.ukmt.org.uk/home/bmo1-1993.pdf">Problem 1</a>.
%H A030466 Michael Penn, <a href="https://www.youtube.com/watch?v=0Hl75Pap6iY">British Mathematics Olympiad 1993 Round 1 Question 1</a>, YouTube, Apr 24, 2020.
%H A030466 Pante Stanica, <a href="https://wcnt.files.wordpress.com/2017/12/wcnt2017-pante.pdf">Squares as concatenation of consecutive integers</a>, Slides, West Coast Number Theory, Dec 17 2017.
%H A030466 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%F A030466 a(n) = A030465(n)*(10^A055642(A030465(n))+1)+1. - _Iain Fox_, Oct 16 2021
%t A030466 fQ[n_] := IntegerQ[Sqrt[n*10^Floor[1 + Log10[n + 1]] + n + 1]]; (* _Robert G. Wilson v_, Dec 27 2017 *)
%o A030466 (PARI) lista(nn) = forstep(n=183, nn, [3, 5, 7, 5, 3, 1, 4, 7, 5, 3, 5, 7, 5, 3, 5, 7, 5, 3, 5, 7, 4, 1], my(s = eval(concat(Str(n), Str(n+1)))); if(issquare(s), print1(s, ", "))) \\ _Iain Fox_, Dec 27 2017
%o A030466 (PARI) eea(x, y) = my(a=max(x,y), b=min(x,y), s=0, so=1, st, r=b, ro=a, rt, q, t); while(r, q=ro\r; rt=r; r=ro-q*r; ro=rt; st=s; s=so-q*s; so=st); t=(ro-so*a)\b; if(x>y, [so, t], [t, so]) \\ Extended Euclidean Algorithm
%o A030466 lista(nn) = my(res=Set(), b, f2, c, s); for(d=3, nn, b=10^d+1; fordiv(b, f, if(f!=1 && f!=b, f2=b/f; if(gcd(f, f2)==1, c=eea(f, f2); if(c[1]<0, s=f*(f2+2*c[1])*f2*(f-2*c[2])+1, s=f*(2*c[1])*f2*(-2*c[2])+1); if(#digits(s)==d*2, res=setunion(res, Set(s))))))); Vec(res) \\ (Will find all values of length nn*2 or shorter) _Iain Fox_, Oct 16 2021
%Y A030466 Cf. A000993, A030465, A030467, A054214, A054215, A054216, A020339, A020340.
%K A030466 nonn,base
%O A030466 1,1
%A A030466 _Patrick De Geest_
%E A030466 a(15)-a(17) from _Arkadiusz Wesolowski_, Apr 02 2014
%E A030466 a(18) from _Iain Fox_, Dec 27 2017