Search: a180416 -id:a180416
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A049416
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Largest number whose square has n digits.
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+10
9
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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a(n) = ceiling(sqrt(10^n)) - 1.
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EXAMPLE
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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.
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,base,easy,nice
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AUTHOR
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Ulrich Schimke (ulrschimke(AT)aol.com)
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 16 2001
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STATUS
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approved
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A167615
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Total number of positive integers below 10^n with 4 positive squares in their representation as sum of squares.
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+10
4
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1, 15, 165, 1665, 16664, 166664, 1666663, 16666663, 166666661, 1666666662, 16666666661, 166666666660, 1666666666661, 16666666666660, 166666666666659, 1666666666666660, 16666666666666658, 166666666666666657, 1666666666666666660, 16666666666666666656
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(1) = 1 since 7 is the only natural number below 10 which is the sum of 4 but no fewer nonzero squares.
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MAPLE
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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;
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A180425
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Number of positive integers below 10^n requiring 3 positive squares in their representation as sum of squares.
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+10
4
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2, 42, 505, 5586, 59308, 616995, 6347878, 64875490, 660104281, 6695709182, 67762820595, 684596704482, 6907026402474, 69611115440126, 700946070114283, 7053023642205904
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OFFSET
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1,1
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A180426
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The number of n-digit numbers requiring 2 nonzero squares in their representation as sum of squares.
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+10
3
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3, 30, 265, 2351, 21062, 191630, 1766955, 16465551, 154749588, 1464326721, 13932672360, 133165432342, 1277568139729, 12295904124627, 118665023703067, 1147922359531155
(list;
graph;
refs;
listen;
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn,more,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A164775
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a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.
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+10
2
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7, 43, 330, 2749, 24028, 216341, 1985459, 18457847, 173229058, 1637624156, 15570512744, 148736628858, 1426306930865, 13722217893214, 132387263219058, 1280309691127436
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listen;
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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