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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: 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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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], 2012-2013.

Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem

Wikipedia, Lagrange's Four-Square 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[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);

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-(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

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 2010-02-07. Luis F.B.A. Alexandre (lfbaa(AT)di.ubi.pt), Feb 08 2010

STATUS

approved

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Last modified November 18 10:33 EST 2017. Contains 294887 sequences.