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Number of numbers m <= n such that 4 equals the first digit after decimal point of square root of n in decimal representation.
12

%I #12 May 17 2024 09:51:07

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

%C For n > 1: if A023961(n)=4 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.

%H Paolo Xausa, <a href="/A111854/b111854.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 2, a(100) = 13, a(1000) = 112, a(10000) = 1030.

%t Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 4] &, 100]] (* _Paolo Xausa_, May 17 2024 *)

%Y Cf. A111850, A111851, A111852, A111853, A111855, A111856, A111857, A111858, A111859, A111894.

%K nonn,base

%O 1,6

%A _Reinhard Zumkeller_, Aug 20 2005