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 1-5 of 5 results.

A049416 Largest number whose square has n digits.

Original entry on oeis.org

3, 9, 31, 99, 316, 999, 3162, 9999, 31622, 99999, 316227, 999999, 3162277, 9999999, 31622776, 99999999, 316227766, 999999999, 3162277660, 9999999999, 31622776601, 99999999999, 316227766016, 999999999999, 3162277660168
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

a(n) + A180416(n) + A180425(n) + A167615(n) = A002283(n).

Examples

			31^2 = 961, but 32^2 = 1024, hence a(3) = 31.
a(4) = 99: 99^2 = 9801 has 4 digits, while 100^2 = 10000 has 5 digits.
		

Crossrefs

Cf. A061433, A049415. Equals A017936 - 1.

Programs

  • Magma
    [Ceiling(Sqrt(10^n))-1: n in [1..30]]; // Vincenzo Librandi, Oct 01 2011
  • Mathematica
    Ceiling[Sqrt[10^Range[40]]-1] (* Harvey P. Dale, Sep 30 2011 *)

Formula

a(n) = ceiling(sqrt(10^n)) - 1.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001

A167615 Total number of positive integers below 10^n with 4 positive squares in their representation as sum of squares.

Original entry on oeis.org

1, 15, 165, 1665, 16664, 166664, 1666663, 16666663, 166666661, 1666666662, 16666666661, 166666666660, 1666666666661, 16666666666660, 166666666666659, 1666666666666660, 16666666666666658, 166666666666666657, 1666666666666666660, 16666666666666666656
Offset: 1

Views

Author

Martin Renner, Jan 18 2011

Keywords

Comments

A049416(n) + A180416(n) + A180425(n) + a(n) = A002283(n).

Examples

			a(1) = 1 since 7 is the only natural number below 10 which is the sum of 4 but no fewer nonzero squares.
		

Crossrefs

Cf. A004215.

Programs

  • Maple
    a:=proc(n)
      local f,s,k;
      f:=(x,y)->ceil(10^y/2^(2*x+3)-7/8):
      s:=0:
      for k from 0 by 1 while not f(k,n)=0 do
        s:=s+f(k,n);
      od:
      return(s);
    end;
  • Mathematica
    a[n_] := Module[{f, s = 0, k}, f[x_, y_] := Ceiling[10^y/2^(2x+3) - 7/8]; For[k = 0, f[k, n] != 0, k++, s += f[k, n]]; Return[s]];
    Array[a, 20] (* Jean-François Alcover, Oct 31 2020, after Maple *)

Formula

a(n) = Sum_{i=0..k} ceiling(10^n/2^(2*i+3) - 7/8) with minimal k for which ceiling(10^n/2^(2*k+3) - 7/8) = 0.

A180425 Number of positive integers below 10^n requiring 3 positive squares in their representation as sum of squares.

Original entry on oeis.org

2, 42, 505, 5586, 59308, 616995, 6347878, 64875490, 660104281, 6695709182, 67762820595, 684596704482, 6907026402474, 69611115440126, 700946070114283, 7053023642205904
Offset: 1

Views

Author

Martin Renner, Jan 19 2011

Keywords

Crossrefs

Formula

a(n) = #{k: A000419(k) < 10^n}.
A049416(n) + A180416(n) + a(n) + A167615(n) = A002283(n).

Extensions

a(6)=616995 by Lars Blomberg, May 03 2011
a(7)-a(10) from Donovan Johnson, Jul 01 2011
a(10) corrected and a(11)-a(16) from Hiroaki Yamanouchi, Jul 13 2014

A180426 The number of n-digit numbers requiring 2 nonzero squares in their representation as sum of squares.

Original entry on oeis.org

3, 30, 265, 2351, 21062, 191630, 1766955, 16465551, 154749588, 1464326721, 13932672360, 133165432342, 1277568139729, 12295904124627, 118665023703067, 1147922359531155
Offset: 1

Views

Author

Martin Renner, Jan 19 2011

Keywords

Comments

A049415(n) + a(n) + A180429(n) + A180347(n) = A052268(n)

Crossrefs

Formula

a(n) = A180416(n)-A180416(n-1) for n>1.

Extensions

a(5)-a(8) from Alois P. Heinz, Jan 20 2011
a(9)-a(10) from Donovan Johnson, Jul 01 2011
a(10) corrected and a(11)-a(16) added by Hiroaki Yamanouchi, Aug 30 2014

A164775 a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

Original entry on oeis.org

7, 43, 330, 2749, 24028, 216341, 1985459, 18457847, 173229058, 1637624156, 15570512744, 148736628858, 1426306930865, 13722217893214, 132387263219058, 1280309691127436
Offset: 1

Views

Author

Eric W. Weisstein, Aug 26 2009

Keywords

Examples

			a(1)=7 since 1 = 0^2 + 1^2, 2 = 1^2 + 1^2, 4 = 0^2 + 2^2, 5 = 1^2 + 2^2, 8 = 2^2 + 2^2, 9 = 0^2 + 3^2, 10 = 1^2 + 3^3.
		

Crossrefs

Formula

a(n) = A180416(n) + ceiling(sqrt(10^n)). - Hiroaki Yamanouchi, Jul 14 2014

Extensions

Offset changed from 0 to 1 by Robert G. Wilson v, Aug 29 2009
a(9) from Eric W. Weisstein, Aug 29 2009
a(10) from Donovan Johnson, Sep 16 2009
a(11)-a(12) from Ant King, May 02 2010
a(11)-a(12) corrected and a(13)-a(16) added by Hiroaki Yamanouchi, Jul 14 2014
Showing 1-5 of 5 results.