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A018851 a(n)^2 is smallest square beginning with n.

%I A018851 #53 Jul 17 2016 11:10:09
%S A018851 0,1,5,6,2,23,8,27,9,3,10,34,11,37,12,39,4,42,43,14,45,46,15,48,49,5,
%T A018851 51,52,17,54,55,56,18,58,59,188,6,61,62,63,20,203,65,66,21,213,68,69,
%U A018851 22,7,71,72,23,73,74,235,75,24,241,77,78,247,25,251,8,81,257,26,83,263,84,267,27
%N A018851 a(n)^2 is smallest square beginning with n.
%C A018851 The following discussion is based on comments from _David A. Corneth_, _Robert Israel_, _N. J. A. Sloane_, and _Chai Wah Wu_. (Start)
%C A018851 As the graph shows, the points belong to various "curves". For each n there is a value d = d(n) such that n*10^d <= a(n)^2 < (n+1)*10^d, and so on this curve, a(n) ~ sqrt(n)*10^(d/2). The values of d(n) are given in A272677.
%C A018851 For a given n, d can range from 0 (if n is a square) to d_max, where it appears that d_max approx. equals 3 + floor( log_10(n/25) ). The successive points where d_max increases are given in A272678, and that entry contains more precise conjectures about the values.
%C A018851 For example, in the range 2600 = A272678(5) <= n < 25317 = A272678(6), d_max is 5. This is the upper curve in the graph that is seen if the "graph" button is clicked, and on this curve a(n) is about sqrt(n)*10^(5/2). (End)
%H A018851 N. J. A. Sloane, <a href="/A018851/b018851.txt">Table of n, a(n) for n = 0..20000</a>
%H A018851 Zak Seidov, <a href="http://zak08.livejournal.com/9621.html">Blog entry</a>
%H A018851 Zak Seidov, <a href="/A018851/a018851.pdf">Blog entry</a> [Cached copy, pdf format, with permission]
%F A018851 a(n) >= sqrt(n), for all n >= 0 with equality when n is a square. - _Derek Orr_, Jul 26 2015
%p A018851 f:= proc(n) local d,m;
%p A018851   for d from 0 do
%p A018851     m:= ceil(sqrt(10^d*n));
%p A018851     if m^2 < 10^d*(n+1) then return m fi
%p A018851   od
%p A018851 end proc:
%p A018851 map(f, [$1..100]); # _Robert Israel_, Jul 26 2015
%o A018851 (PARI) a(n)=k=1; while(k,d=digits(k^2); D=digits(n); if(#D<=#d,c=1; for(i=1,#D,if(D[i]!=d[i],c=0;break));if(c,return(k)));k++)
%o A018851 vector(100,n,a(n)) \\ _Derek Orr_, Jul 26 2015
%Y A018851 Cf. A018796 (the squares), A272677, A272678.
%Y A018851 A265432 is a more complicated sequence in the same spirit.
%K A018851 nonn,base,look
%O A018851 0,3
%A A018851 _David W. Wilson_
%E A018851 Added initial 0. - _N. J. A. Sloane_, May 21 2016
%E A018851 Comments revised by _N. J. A. Sloane_, Jul 17 2016