A180968 The only integers that cannot be partitioned into a sum of six positive squares.

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19

Offset: 1

Author

Ant King, Sep 30 2010

Keywords

• fini
• full
• nonn

Comments

From R. J. Mathar, Sep 11 2012: (Start)

Not the sum of 7 positive squares: 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 20.

Not the sum of 8 positive squares: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 21.

Not the sum of 9 positive squares: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 19, 22.

Not the sum of 10 positive squares: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 20, 23. (End)

Examples

			As the sixth integer which cannot be partitioned into a sum of six positive squares is 7, we have a(6)=7.
		

References

• Dubouis, E.; L'Interm. des math., vol. 18, (1911), pp. 55-56, 224-225.
• Grosswald, E.; Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), pp.73-74.

Links

• Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.

Crossrefs

Cf. A047701 (not the sum of 5 squares)

Programs

Mathematica

s=6;B={1,2,4,5,7,10,13}; Union[Range[s-1],s+B]//Sort

Formula

Let B be the set of integers {1,2,4,5,7,10,13}. Then, for s>=6, every integer can be partitioned into a sum of s positive squares except for 1,2,...,s-1 and s+b where b is a member of the set B [Dubouis].