A023961 -id:A023961 - cp's OEIS frontend

A034096 Fractional part of square root of n starts with digit 0 (squares excluded).

26, 37, 50, 65, 82, 101, 102, 122, 123, 145, 146, 170, 171, 197, 198, 226, 227, 228, 257, 258, 259, 290, 291, 292, 325, 326, 327, 362, 363, 364, 401, 402, 403, 404, 442, 443, 444, 445, 485, 486, 487, 488, 530, 531, 532, 533, 577, 578, 579, 580, 626, 627

Offset: 1

Patrick De Geest, Sep 15 1998

Given n > 4, n^2 + 1 is in the sequence. In fact, as n gets larger, more and more numbers just above n^2 are also in the sequence. For a particular n, the integers between n^2 and (n + 1/10)^2 are in this sequence. - Alonso del Arte, Mar 16 2019

			sqrt(145) = 12.041594578792295..., so 145 is in the sequence.
sqrt(146) = 12.083045973594572..., so 146 is also in the sequence.
sqrt(147) = 12.124355652982141..., so 147 is not in the sequence.

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

Cf. A023961, A034106.

A034096 := proc(n) option remember: local k,rt: if(n=1)then return 26: else k:=procname(n-1)+1: do rt:=sqrt(k): if(not frac(rt)=0 and floor(10*rt) mod 10 = 0)then return k: fi: k:=k+1: od: fi: end: seq(A034096(n), n=1..50); # Nathaniel Johnston, May 04 2011
seq(seq(x, x=floor(n^2) +1 .. ceil((n+1/10)^2)-1),n=1..100); # Robert Israel, Sep 21 2015

zdQ[n_] := Module[{c = Sqrt[n], sr, i, l}, sr = RealDigits[c, 10, 5]; i = Last[sr] + 1; l = First[sr]; l[[i]] == 0 && !IntegerQ[c]]; Select[Range[700], zdQ] (* Harvey P. Dale, Oct 10 2011 *)
Flatten[Table[Range[n^2 + 1, Floor[(n + 1/10)^2]], {n, 25}]] (* Alonso del Arte, Mar 16 2019 *)

isok(n) = !issquare(n) && !(floor(10*sqrt(n)) % 10); \\ Michel Marcus, Sep 21 2015

is(n)=my(s=sqrtint(n),s2=s^2); s2+s\5 >= n && s2 < n \\ Charles R Greathouse IV, Sep 07 2022

list(lim)=my(v=List(),s=sqrtint(lim\=1)); for(n=5,s-1, for(i=n^2+1,n^2+n\5, listput(v,i))); for(i=s^2+1,min(s^2+s\5,lim), listput(v,i)); Vec(v) \\ Charles R Greathouse IV, Sep 08 2022

A023961(a(n)) = 0. - Michel Marcus, Sep 21 2015

a(n) = 10n + O(sqrt(n)). - Charles R Greathouse IV, Sep 08 2022

Name clarified by Michel Marcus, Sep 21 2015

A111862 Second digit after decimal point of square root of n in decimal representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 20 2005

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

Cf. A023961, A111890, A111891, A111892, A111893, A111894, A111895, A111896, A111897, A111898, A111899.

A111862 := proc(n) return floor(100*sqrt(n)) mod 10: end: seq(A111862(n), n=1..100); # Nathaniel Johnston, May 04 2011

Mod[Floor[100*Sqrt[Range[100]]], 10] (* Paolo Xausa, May 17 2024 *)

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 *)

A111858 Number of numbers m <= n such that 8 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, 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, 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, 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

Offset: 1

Reinhard Zumkeller, Aug 20 2005

			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.

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

Cf. A023961, A111850, A111851, A111852, A111853, A111854, A111855, A111856, A111857, A111859, A111898.

Accumulate[Array[Boole[Mod[Floor[10*Sqrt[#]], 10] == 8] &, 100]] (* Paolo Xausa, May 17 2024 *)

For n > 1: if A023961(n) = 8 then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1).

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

cp's OEIS Frontend

A034096 Fractional part of square root of n starts with digit 0 (squares excluded).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A111862 Second digit after decimal point of square root of n in decimal representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica