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A111854 Number of numbers m <= n such that 4 equals the first digit after decimal point of square root of n in decimal representation.

%I A111854 #12 May 17 2024 09:51:07
%S A111854 0,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,
%T A111854 5,5,5,5,5,5,6,7,7,7,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9,9,9,9,9,9,9,9,9,9,
%U A111854 9,9,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,13,13,13,13
%N A111854 Number of numbers m <= n such that 4 equals the first digit after decimal point of square root of n in decimal representation.
%C A111854 For n > 1: if A023961(n)=4 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).
%C A111854 Lim_{n->infinity} a(n)/n = 1/10.
%D A111854 G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 4, Sect. 4, Problem 178.
%H A111854 Paolo Xausa, <a href="/A111854/b111854.txt">Table of n, a(n) for n = 1..10000</a>
%e A111854 a(10) = 2, a(100) = 13, a(1000) = 112, a(10000) = 1030.
%t A111854 Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 4] &, 100]] (* _Paolo Xausa_, May 17 2024 *)
%Y A111854 Cf. A111850, A111851, A111852, A111853, A111855, A111856, A111857, A111858, A111859, A111894.
%K A111854 nonn,base
%O A111854 1,6
%A A111854 _Reinhard Zumkeller_, Aug 20 2005