login
Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.
11

%I #10 Dec 24 2019 21:41:25

%S 1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,6,6,6,

%T 6,7,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,

%U 10,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,14,14,14,14,14

%N Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.

%C For n > 1: if A023961(n)=0 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

%C Lim_{n->infinity} a(n)/n = 1/10.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.

%e a(10) = 3, a(100) = 15, a(1000) = 118, a(10000) = 1050.

%t zd[n_]:=Module[{c=RealDigits[Sqrt[n],10,10],f},f=Last[c]+1;If[First[c][[f]]==0,1,0]]; Accumulate[Array[zd,90]] (* _Harvey P. Dale_, Feb 01 2012 *)

%Y Cf. A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111859, A111890.

%K nonn,base

%O 1,4

%A _Reinhard Zumkeller_, Aug 20 2005