A111853 Number of numbers m <= n such that 3 equals the first digit after decimal point of square root of n in decimal representation.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 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, 7, 7, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Author

Reinhard Zumkeller, Aug 20 2005

Keywords

• nonn
• base

Comments

For n > 1: if A023961(n)=3 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

Examples

			a(10) = 0, a(100) = 9, a(1000) = 99, a(10000) = 990.
		

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A111850, A111851, A111852, A111854, A111855, A111856, A111857, A111858, A111859, A111893.

Programs

Mathematica

fddpQ[n_]:=Module[{a,b},{a,b}=RealDigits[Surd[n,2],10,10];a[[b+1]] == 3]; Accumulate[Table[If[fddpQ[n],1,0],{n,110}]] (* Harvey P. Dale, Feb 06 2019 *)
Accumulate[Table[If[NumberDigit[Sqrt[n],-1]==3,1,0],{n,110}]] (* Harvey P. Dale, Aug 04 2025 *)