cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A384923 a(n) is the smallest number of leading significant digits of the square root of the n-th nonsquare that includes all decimal digits.

Original entry on oeis.org

19, 23, 37, 39, 45, 36, 27, 17, 25, 15, 36, 19, 20, 36, 25, 37, 28, 13, 27, 52, 39, 17, 38, 27, 26, 17, 23, 24, 37, 19, 25, 26, 26, 41, 58, 57, 25, 12, 25, 22, 24, 19, 33, 48, 23, 41, 49, 23, 32, 32, 23, 30, 19, 17, 31, 27, 24, 47, 24, 26, 18, 22, 19, 48, 31, 22
Offset: 1

Views

Author

Felix Huber, Jun 26 2025

Keywords

Comments

Squares are excluded by definition because a(n) would only exist for positive integers s that include all decimal digits. The smallest square s^2 for which a(n) would exist is 1023456789^2 = 1047463798950190521.

Examples

			The leading 19 significant digits of sqrt(2) are [1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0, 9, 5, 0, 4, 8]. These digits include all decimal digits, with the digit '8' appearing for the first time at position 19. Since 2 is the first nonsquare, it follows that a(1) = 19.

Links

Crossrefs

Cf. A000037, A000196, A003285, A061845, A113507, A384924.

Programs

  • Maple
    A384923:=proc(n)
    local m,b,k;
    m:=n+floor(1/2+sqrt(n));
    b:=floor(log10(sqrt(m)));
    k:=9-b;
    while nops(convert(ListTools:-Reverse(convert(floor(10^k*sqrt(m)),'base',10)),set))<10 do
        k:=k+1
    od;
    return k+b+1
end proc;
seq(A384923(n),n=1..66);
  • Python
    from itertools import count
from math import isqrt
def A384923(n):
    m = n+(k:=isqrt(n))+(n>k*(k+1))
    return 1+next(n for n in count(9) if len(set(str(isqrt(10**(n<<1)*m))))==10) # Chai Wah Wu, Jul 01 2025

Formula

a(n) >= max(10, A384924(n)).
a(A113507(k) - floor(sqrt(A113507(k)))) = 10 for positive integers k.

A384924 a(n) is the position of the first occurrence of the digit 0 among the leading significant decimal digits of the square root of the n-th nonsquare.

Original entry on oeis.org

14, 5, 5, 17, 11, 16, 10, 10, 6, 3, 36, 12, 6, 7, 13, 37, 16, 4, 26, 52, 2, 12, 6, 9, 11, 13, 16, 14, 4, 5, 2, 8, 18, 10, 3, 4, 12, 10, 3, 20, 9, 6, 2, 48, 6, 4, 49, 11, 32, 13, 9, 15, 19, 4, 5, 21, 2, 5, 24, 17, 3, 6, 19, 16, 5, 3, 4, 11, 17, 7, 19, 9, 2, 4, 16
Offset: 1

Views

Author

Felix Huber, Jun 26 2025

Keywords

Examples

			The leading 14 significant digits of sqrt(2) are [1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3, 7, 3, 0], with the digit '0' appearing for the first time at position 14. Since 2 is the first nonsquare, it follows that a(1) = 14.

Links

Crossrefs

Cf. A000037, A000196, A003285, A113507, A384923.

Programs

  • Maple
    A384924:=proc(n)
    local m,b,k;
    m:=n+floor(1/2+sqrt(n));
    b:=floor(log10(sqrt(m)));
    k:=1-b;
    while not member(0,ListTools:-Reverse(convert(floor(10^k*sqrt(m)),'base',10))) do
        k:=k+1
    od;
    return k+b+1
end proc;
seq(A384924(n),n=1..75);
  • Mathematica
    b[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]);a[n_]:=Position[RealDigits[N[Sqrt[b[n]],100]][[1]],0][[1]];Array[a,75]//Flatten (* Increase precision for n>23000 *) (* James C. McMahon, Jul 05 2025 *)
  • Python
    from itertools import count
from math import isqrt
def A384924(n):
    m = n+(k:=isqrt(n))+(n>k*(k+1))
    return 1+next(n for n in count(1) if not isqrt(10**(n<<1)*m)%10) # Chai Wah Wu, Jul 01 2025

Formula

2 <= a(n) <= A384923(n).

A119517 The first 10 digits of the cube root of n contain the digits 0-9.

Original entry on oeis.org

2017, 3053, 9950, 15139, 15533, 18357, 24214, 24424, 31457, 32654, 39605, 46705, 47776, 57692, 60448, 65839, 65854, 66999, 67405, 68512, 70239, 73985, 74283, 74493, 77913, 79600, 82431, 83311, 84467, 91571, 95557
Offset: 1

Views

Author

Cino Hilliard, May 27 2006

Keywords

Comments

i = 2 produces A113507 in the PARI script.

Examples

			n=9950. n^(1/3) = 21.50837964..., so 9950 is the third entry.

Crossrefs

Cf. A113507.

Programs

  • Mathematica
    Select[Range[100000],Sort[RealDigits[Surd[#,3],10,10][[1]]]==Range[0,9]&] (* Harvey P. Dale, Jan 22 2013 *)
  • PARI
    \\ The first 10 digits of i-th root of x contain all of the digits 0-9. rootdigits(n,i) = { local(f,x,y,a,d,s); for(x=2,n, f=[0,0,0,0,0,0,0,0,0,0]; s=0; y=(x^(1/i))*10^9; a=Vec(Str(y)); for(d=1,10, k=eval(a[d]); if(k==0,k=10); f[k]=1; ); for(j=1,10,s+=f[j]); if(s==10,print1(x",")); ) }

A119519 The first 10 digits of the fourth root of n contain the digits 0-9.

Original entry on oeis.org

6654, 14311, 14422, 14505, 24364, 25646, 33421, 35833, 36759, 36870, 37112, 37628, 41108, 42606, 45886, 46453, 46729, 47183, 49698, 50064, 56023, 66932, 69520, 70236, 70367, 71443, 71898, 73005, 73676, 74488, 74972, 75464, 78872, 82066
Offset: 1

Views

Author

Cino Hilliard, May 27 2006

Keywords

Examples

			n=6654. n(1/4) = 9.031724865..., so 6654 is the first entry.

Crossrefs

Cf. A113507.

Programs

  • Mathematica
    okQ[n_]:=With[{ptrn=Table[1,{10}]},Module[{rd10=RealDigits[Power[n, (4)^-1],10,10][[1]]},DigitCount[FromDigits[rd10]]==ptrn]]; Select[Range[90000],okQ]  (* Harvey P. Dale, Jan 21 2011 *)
  • PARI
    \\ The first 10 digits of i-th root of x contain all of the digits 0-9. rootdigits(n,i) = { local(f,x,y,a,d,s); for(x=2,n, f=[0,0,0,0,0,0,0,0,0,0]; s=0; y=(x^(1/i))*10^9; a=Vec(Str(y)); for(d=1,10, k=eval(a[d]); if(k==0,k=10); f[k]=1; ); for(j=1,10,s+=f[j]); if(s==10,print1(x",")); ) }
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.