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 A018851 a(n)^2 is smallest square beginning with n. 21
 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, 51, 52, 17, 54, 55, 56, 18, 58, 59, 188, 6, 61, 62, 63, 20, 203, 65, 66, 21, 213, 68, 69, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The following discussion is based on comments from David A. Corneth, Robert Israel, N. J. A. Sloane, and Chai Wah Wu. (Start) 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. 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. 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) LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..20000 Zak Seidov, Blog entry Zak Seidov, Blog entry [Cached copy, pdf format, with permission] FORMULA a(n) >= sqrt(n), for all n >= 0 with equality when n is a square. - Derek Orr, Jul 26 2015 MAPLE f:= proc(n) local d, m;   for d from 0 do     m:= ceil(sqrt(10^d*n));     if m^2 < 10^d*(n+1) then return m fi   od end proc: map(f, [\$1..100]); # Robert Israel, Jul 26 2015 PROG (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++) vector(100, n, a(n)) \\ Derek Orr, Jul 26 2015 CROSSREFS Cf. A018796 (the squares), A272677, A272678. A265432 is a more complicated sequence in the same spirit. Sequence in context: A201332 A049253 A072733 * A260463 A226533 A011499 Adjacent sequences:  A018848 A018849 A018850 * A018852 A018853 A018854 KEYWORD nonn,base,look AUTHOR EXTENSIONS Added initial 0. - N. J. A. Sloane, May 21 2016 Comments revised by N. J. A. Sloane, Jul 17 2016 STATUS approved

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Last modified May 14 16:23 EDT 2021. Contains 343884 sequences. (Running on oeis4.)