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Search: a180416 -id:a180416
Displaying 1-5 of 5 results found.
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A049416
Largest number whose square has n digits.
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) + A180416(n) + A180425(n) + A167615(n) = A002283(n).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = ceiling(sqrt(10^n)) - 1.
EXAMPLE
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.
MATHEMATICA
Ceiling[Sqrt[10^Range[40]]-1] (* Harvey P. Dale, Sep 30 2011 *)
PROG
(Magma) [Ceiling(Sqrt(10^n))-1: n in [1..30]]; // Vincenzo Librandi, Oct 01 2011
CROSSREFS
Cf. A061433, A049415. Equals A017936 - 1.
KEYWORD
nonn,base,easy,nice
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
STATUS
approved
A167615
Total number of positive integers below 10^n with 4 positive squares in their representation as sum of squares.
+10
4
1, 15, 165, 1665, 16664, 166664, 1666663, 16666663, 166666661, 1666666662, 16666666661, 166666666660, 1666666666661, 16666666666660, 166666666666659, 1666666666666660, 16666666666666658, 166666666666666657, 1666666666666666660, 16666666666666666656
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A049416(n) + A180416(n) + A180425(n) + a(n) = A002283(n).
LINKS
Table of n, a(n) for n=1..20.
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem.
Eric Weisstein's World of Mathematics, Sum of Squares Function.
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.
EXAMPLE
a(1) = 1 since 7 is the only natural number below 10 which is the sum of 4 but no fewer nonzero squares.
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 *)
CROSSREFS
Cf. A004215.
KEYWORD
nonn
AUTHOR
Martin Renner, Jan 18 2011
STATUS
approved
A180425
Number of positive integers below 10^n requiring 3 positive squares in their representation as sum of squares.
+10
4
2, 42, 505, 5586, 59308, 616995, 6347878, 64875490, 660104281, 6695709182, 67762820595, 684596704482, 6907026402474, 69611115440126, 700946070114283, 7053023642205904
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..16.
Eric W. Weisstein, MathWorld -- Lagrange's Four-Square Theorem.
Eric W. Weisstein, MathWorld -- Sum of Squares Function.
FORMULA
a(n) = #{k: A000419(k) < 10^n}.
A049416(n) + A180416(n) + a(n) + A167615(n) = A002283(n).
CROSSREFS
Cf. A000419, A049416, A180416, A167615, A002283.
KEYWORD
nonn,more
AUTHOR
Martin Renner, Jan 19 2011
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
STATUS
approved
A180426
The number of n-digit numbers requiring 2 nonzero squares in their representation as sum of squares.
+10
3
3, 30, 265, 2351, 21062, 191630, 1766955, 16465551, 154749588, 1464326721, 13932672360, 133165432342, 1277568139729, 12295904124627, 118665023703067, 1147922359531155
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A049415(n) + a(n) + A180429(n) + A180347(n) = A052268(n)
LINKS
Table of n, a(n) for n=1..16.
Eric W. Weisstein: MathWorld -- Lagrange's Four-Square Theorem.
Eric W. Weisstein: MathWorld -- Sum of Squares Function.
FORMULA
a(n) = A180416(n)-A180416(n-1) for n>1.
CROSSREFS
Cf. A000404, A180416.
KEYWORD
nonn,more,base
AUTHOR
Martin Renner, Jan 19 2011
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
STATUS
approved
A164775
a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.
+10
2
7, 43, 330, 2749, 24028, 216341, 1985459, 18457847, 173229058, 1637624156, 15570512744, 148736628858, 1426306930865, 13722217893214, 132387263219058, 1280309691127436
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..16.
Peter Shiu, Counting Sums of Two Squares: The Meissel-Lehmer Method, Mathematics of Computation 47:175 (July 1986), pp. 351-360. [Beware errors.]
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant
FORMULA
a(n) = A180416(n) + ceiling(sqrt(10^n)). - Hiroaki Yamanouchi, Jul 14 2014
EXAMPLE
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
Cf. A000050, A001481, A064533, A227158, A275649, A275650.
KEYWORD
nonn,hard,more
AUTHOR
Eric W. Weisstein, Aug 26 2009
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
STATUS
approved
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