login
Number of numbers m <= n such that 5 equals the first digit after decimal point of square root of n in decimal representation.
12

%I #14 May 17 2024 15:29:58

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,

%T 2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,

%U 4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7

%N Number of numbers m <= n such that 5 equals the first digit after decimal point of square root of n in decimal representation.

%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="/A111855/b111855.txt">Table of n, a(n) for n = 1..10000</a>

%F For n > 1: if A023961(n)=5 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

%F Limit_{n->oo} a(n)/n = 1/10.

%e a(10) = 0, a(100) = 7, a(1000) = 93, a(10000) = 970.

%t Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 5] &, 100]] (* _Paolo Xausa_, May 17 2024 *)

%Y Cf. A111850, A111851, A111852, A111853, A111854, A111856, A111857, A111858, A111859, A111895.

%K nonn,base

%O 1,31

%A _Reinhard Zumkeller_, Aug 20 2005