

A180968


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


2



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

1,2


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)


REFERENCES

Dubouis, E.; L'Interm. des math., vol. 18, (1911), pp. 5556, 224225.
Grosswald, E.; Representation of Integers as Sums of Squares, SpringerVerlag, New York Inc., (1985), pp.7374.


LINKS

Table of n, a(n) for n=1..12.
Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 1018.


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,...,s1 and s+b where b is a member of the set B [Dubouis].


EXAMPLE

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


MATHEMATICA

s=6; B={1, 2, 4, 5, 7, 10, 13}; Union[Range[s1], s+B]//Sort


CROSSREFS

Cf. A047701 (not the sum of 5 squares)
Sequence in context: A025199 A213858 A277992 * A191847 A321290 A274337
Adjacent sequences: A180965 A180966 A180967 * A180969 A180970 A180971


KEYWORD

fini,full,nonn


AUTHOR

Ant King, Sep 30 2010


STATUS

approved



