A111858 -id:A111858 - cp's OEIS frontend

A023961 First digit after decimal point of square root of n.

0, 4, 7, 0, 2, 4, 6, 8, 0, 1, 3, 4, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 0, 0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 8, 9, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 0, 0, 0, 1, 1, 2

Offset: 1

N. J. A. Sloane, Olivier Gérard

When n is a square, a(n) is equal to 0, but the converse is not true, see A034096. - Michel Marcus, Sep 21 2015

			sqrt(1) = 1.00000000... hence a(1) = 0.
sqrt(2) = 1.41421356... hence a(2) = 4.
sqrt(3) = 1.73205080... hence a(3) = 7.
sqrt(4) = 2.00000000... hence a(4) = 0.
Note that 26 = 2 * 13 and sqrt(26) = 5.09901951... so a(26) = 0 even though 26 is not a perfect square.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A111862, A111850, A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111859.

A023961 := proc(n) return floor(10*sqrt(n)) mod 10: end: seq(A023961(n),n=1..100); # Nathaniel Johnston, May 04 2011

Array[Function[n, RealDigits[N[Power[n, 1/2], 10], 10] // (#[[1, #[[2]] + 1]]) &], 110]
fd[n_] := Module[{rd = RealDigits[Sqrt[n], 10, 10]}, First[rd][[Last[rd] + 1]]]; Array[fd, 90] (* Harvey P. Dale, Jan 16 2014 *)

a(n) = floor(10*sqrt(n)) % 10; \\ Michel Marcus, Sep 21 2015

a(n) = A010879(A000196(100*n)). - Robert Israel, Jul 30 2015

A111851 Number of numbers m <= n such that 1 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 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, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 11

Offset: 1

Reinhard Zumkeller, Aug 20 2005

			a(10) = 1, a(100) = 9, a(1000) = 99, a(10000) = 990.

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A111850, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111859, A111891.

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 1] &, 100]] (* Paolo Xausa, May 17 2024 *)

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

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

A111852 Number of numbers m <= n such that 2 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 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, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 9, 10

Offset: 1

Reinhard Zumkeller, Aug 20 2005

For n > 1: if A023961(n)=2 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

			a(10) = 1, a(100) = 9, a(1000) = 99, a(10000) = 990.

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A111850, A111851, A111853, A111854, A111855, A111856, A111857, A111858, A111859, A111892.

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 2] &, 100]] (* Paolo Xausa, May 17 2024 *)

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

Original entry on oeis.org

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

Reinhard Zumkeller, Aug 20 2005

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.

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

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

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

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

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 *)

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

Original entry on oeis.org

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, 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, 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

Offset: 1

Reinhard Zumkeller, Aug 20 2005

For n > 1: if A023961(n)=4 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

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

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 4] &, 100]] (* Paolo Xausa, May 17 2024 *)

A111855 Number of numbers m <= n such that 5 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

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, 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, 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

Offset: 1

Reinhard Zumkeller, Aug 20 2005

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

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 5] &, 100]] (* Paolo Xausa, May 17 2024 *)

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

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

A111856 Number of numbers m <= n such that 6 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 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, 6, 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, 10, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Reinhard Zumkeller, Aug 20 2005

			a(10) = 1, a(100) = 11, a(1000) = 101, a(10000) = 1010.

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A111850, A111851, A111852, A111853, A111854, A111855, A111857, A111858, A111859, A111896.

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 6] &, 100]] (* Paolo Xausa, May 17 2024 *)

For n > 1: if A023961(n)=6 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

A111857 Number of numbers m <= n such that 7 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 10, 11, 11, 11, 11, 11, 11

Offset: 1

Reinhard Zumkeller, Aug 20 2005

For n > 1: if A023961(n)=7 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

			a(10) = 1, a(100) = 11, a(1000) = 99, a(10000) = 1010.

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

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

Cf. A111850, A111851, A111852, A111853, A111854, A111855, A111856, A111858, A111859, A111897.

Accumulate[Table[If[NumberDigit[Sqrt[n],-1]==7,1,0],{n,120}]] (* Harvey P. Dale, Apr 20 2022 *)

A111859 Number of numbers m <= n such that 9 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 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, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Reinhard Zumkeller, Aug 20 2005

			a(10) = 0, a(100) = 5, a(1000) = 81, a(10000) = 950.

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A023961, A111850, A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111858, A111899.

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 9] &, 100]] (* Paolo Xausa, May 17 2024 *)

For n > 1: if A023961(n) = 9 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

A111850 Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

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, 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, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14

Offset: 1

Reinhard Zumkeller, Aug 20 2005

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

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

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

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

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

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 *)

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A023961 First digit after decimal point of square root of n.

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A111851 Number of numbers m <= n such that 1 equals the first digit after decimal point of square root of n in decimal representation.

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A111852 Number of numbers m <= n such that 2 equals the first digit after decimal point of square root of n in decimal representation.

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

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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.

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A111855 Number of numbers m <= n such that 5 equals the first digit after decimal point of square root of n in decimal representation.

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A111856 Number of numbers m <= n such that 6 equals the first digit after decimal point of square root of n in decimal representation.

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