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.

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

Offset: 1

Author

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

Keywords

• nonn

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

Examples

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

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

Programs

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

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 *)

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