

A112687


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: 1find the largest square s smaller than n; 2n=ns 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.


2



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

1,1


COMMENTS

Found while writing a program to decompose integers as sums of four square numbers (following Lagrange's FourSquare Theorem).
Question: does the sum of the reciprocals of the numbers in this sequence converge?  J. Lowell, May 03 2014
Answer: this series is divergent.  Thomas Ordowski, May 22 2016
Numbers n such that A053610(n) > 4.  Thomas Ordowski, May 22 2016


LINKS

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
E. J. Ionascu, Equilateral triangles in Z^4, arXiv:1209.0147 [math.NT], 20122013.
Eric Weisstein's World of Mathematics, Lagrange's FourSquare Theorem
Wikipedia, Lagrange's FourSquare Theorem


EXAMPLE

23 is the first number that cannot be decomposed because 16+4+1+1 falls short by one.


MATHEMATICA

f1[x_]:=Floor[Sqrt[x]];
f2[x_]:=Floor[Sqrt[xf1[x]^2]];
f3[x_]:=Floor[Sqrt[xf1[x]^2f2[x]^2]];
f4[x_]:=Floor[Sqrt[xf1[x]^2f2[x]^2f3[x]^2]];
Err[n_]:=n(f1[n]^2+f2[n]^2+f3[n]^2+f4[n]^2);
Select[Table[n, {n, 0, 5000}], Err[#]!=0&] (* Enrique Pérez Herrero, Dec 19 2013 *)


PROG

(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(j1)*(j1); i=i+1; end; if(n~=0) disp(a); end; end; % Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010


CROSSREFS

Sequence in context: A070664 A243907 A096083 * A107282 A039410 A043233
Adjacent sequences: A112684 A112685 A112686 * A112688 A112689 A112690


KEYWORD

nonn


AUTHOR

Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Dec 31 2005


EXTENSIONS

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 20100207. Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010


STATUS

approved



