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A112687
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Numbers n that cannot be decomposed into the sum of at most 4 square numbers when using the following algorithm: Repeat the following 2 steps 4 times: 1-find the largest square s smaller than n; 2-n=n-s Numbers that can be decomposed yield final values of n=0. The sequence presented is of those numbers where n is not 0 when this algorithm ends.
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2
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23, 32, 43, 48, 56, 61, 71, 76, 79, 88, 93, 96, 107, 112, 115, 119, 128, 133, 136, 140, 143, 151, 156, 159, 163, 166, 167, 176, 181, 184, 188, 191, 192, 203, 208, 211, 215, 218, 219, 224, 232, 237, 240, 244, 247, 248, 253, 263, 268, 271, 275, 278, 279, 284
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OFFSET
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1,1
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COMMENTS
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Found while writing a program to decompose integers as sums of four square numbers (following Lagrange's Four-Square Theorem).
Question: does the sum of the reciprocals of the numbers in this sequence converge? - J. Lowell, May 03 2014
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LINKS
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EXAMPLE
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23 is the first number that cannot be decomposed because 16+4+1+1 falls short by one.
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MATHEMATICA
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f1[x_]:=Floor[Sqrt[x]];
f2[x_]:=Floor[Sqrt[x-f1[x]^2]];
f3[x_]:=Floor[Sqrt[x-f1[x]^2-f2[x]^2]];
f4[x_]:=Floor[Sqrt[x-f1[x]^2-f2[x]^2-f3[x]^2]];
Err[n_]:=n-(f1[n]^2+f2[n]^2+f3[n]^2+f4[n]^2);
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PROG
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(MATLAB) for n=1:312 a=n; i=1; while(i<5 & n~=0) j=1; while(j*j<=n) j=j+1; end; n=n-(j-1)*(j-1); i=i+1; end; if(n~=0) disp(a); end; end; % Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Dec 31 2005
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EXTENSIONS
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Included terms where the final value of n is larger than 1. The fact that some terms might be missing was noted by Alonso del Arte on 2010-02-07. Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010
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STATUS
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approved
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