A023961 -id:A023961 - cp's OEIS frontend

A111859 Number of numbers m <= n such that 9 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, 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 *)

A343739 a(n) is the last digit to appear in sqrt(n) (or -1 if n is a square).

Original entry on oeis.org

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

Offset: 1

Jon E. Schoenfield, Jul 05 2021

For the digit position in sqrt(n) at which the digit a(n) first appears, see A343740.

			a(2)=8 because 8 is the last digit to appear in sqrt(2) = 1.414213562373095048...;
a(24)=0 because 0 is the last digit to appear in sqrt(24) = 4.898979485566356196394568149411782783931894961313340...

Cf. A023961, A343740, A343741, A343742.

Table[If[IntegerQ@ Sqrt@ n, -1, Function[s, FirstPosition[#, Max@ #][[1]] - 1 &@ Array[FirstPosition[s, #][[1]] &, 10, 0]]@ RealDigits[Sqrt[n], 10, 120][[1]]], {n, 84}] (* Michael De Vlieger, Jul 06 2021 *)

a(100^q*n) = a(n), q > 0. - Bernard Schott, Jul 24 2021

A343740 a(n) is the digit position of the first appearance of the last digit to appear in sqrt(n) (or -1 if n is a square).

Original entry on oeis.org

-1, 19, 23, -1, 37, 39, 45, 36, -1, 27, 17, 25, 15, 36, 19, -1, 20, 36, 25, 37, 28, 13, 27, 52, -1, 39, 17, 38, 27, 26, 17, 23, 24, 37, 19, -1, 25, 26, 26, 41, 58, 57, 25, 12, 25, 22, 24, 19, -1, 33, 48, 23, 41, 49, 23, 32, 32, 23, 30, 19, 17, 31, 27, -1, 24

Offset: 1

Jon E. Schoenfield, Jul 05 2021

A343739(n) is the last digit to appear in the decimal expansion of sqrt(n) (or -1 if n is a square), so a(n) is the digit position of the first appearance of the digit A343739(n) in sqrt(n).

(The first digit of sqrt(n) is counted as digit position 1; the decimal point is disregarded.)

			a(2)=19 because A343739(2)=8 and the first appearance of an 8 in sqrt(2) = 1.414213562373095048... is at the 19th digit;
a(24)=52 because A343739(24)=0 and the first appearance of a 0 in sqrt(24) = 4.898979485566356196394568149411782783931894961313340... is at the 52nd digit.

Cf. A023961, A343739, A343741, A343742.

Table[If[IntegerQ@ Sqrt@ n, -1, Function[s, Max@ Array[FirstPosition[s, #][[1]] &, 10, 0]]@ RealDigits[Sqrt[n], 10, 120][[1]]], {n, 65}] (* Michael De Vlieger, Jul 06 2021 *)

a(100^q*n) = a(n), q > 0. - Bernard Schott, Jul 29 2021

A037022 Triangle in which row n has the first n digits of sqrt(n) (truncated).

Original entry on oeis.org

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

Offset: 1

Jonas Persson (jptmp(AT)hotmail.com), N. J. A. Sloane

			Triangle starts:
  1;
  1, 4;
  1, 7, 3;
  2, 0, 0, 0;
  2, 2, 3, 6, 0;
  2, 4, 4, 9, 4, 8;
  2, 6, 4, 5, 7, 5, 1;
  ...

T. D. Noe, Rows n=1..100 of triangle, flattened

Cf. A000196 (first column), A023961 (second column), A037023.

row[n_] := RealDigits[Sqrt[n], 10, n][[1]]; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 03 2016 *)

A328819 Third digit after decimal point of square root of n.

Original entry on oeis.org

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

Offset: 1

Maxim Skorohodov, Oct 28 2019

			sqrt(21) = 4.58257569..., so a(21) = 2.

Maxim Skorohodov, Table of n, a(n) for n = 1..10000

Cf. A023961 (1st digit), A111862 (2nd digit).

Cf. A000196, A010879.

a(n) = floor(10^3*sqrt(n)) % 10; \\ Jinyuan Wang, Nov 03 2019

a(n) = A010879(A000196(1000000*n)).

A328820 Fourth digit after decimal point of square root of n.

Original entry on oeis.org

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

Offset: 1

Maxim Skorohodov, Oct 28 2019

			sqrt(5) = 2.23606798..., so a(5) = 0.

Maxim Skorohodov, Table of n, a(n) for n = 1..10000

Cf. A111862, A023961.

a(n) = A010879(A000196(100000000*n)).

cp's OEIS Frontend

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

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

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A343739 a(n) is the last digit to appear in sqrt(n) (or -1 if n is a square).

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A343740 a(n) is the digit position of the first appearance of the last digit to appear in sqrt(n) (or -1 if n is a square).

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A037022 Triangle in which row n has the first n digits of sqrt(n) (truncated).

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A328819 Third digit after decimal point of square root of n.

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A328820 Fourth digit after decimal point of square root of n.

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