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A180968
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The only integers that cannot be partitioned into a sum of six positive squares.
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3
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1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 16, 19
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OFFSET
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1,2
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18.
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FORMULA
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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].
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EXAMPLE
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As the sixth integer which cannot be partitioned into a sum of six positive squares is 7, we have a(6)=7.
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MATHEMATICA
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s=6; B={1, 2, 4, 5, 7, 10, 13}; Union[Range[s-1], s+B]//Sort
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CROSSREFS
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Cf. A047701 (not the sum of 5 squares)
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KEYWORD
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fini,full,nonn
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AUTHOR
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STATUS
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approved
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