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A263765
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Minimum number of squares necessary to write n as a sum or difference of squares.
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1
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1, 1, 2, 2, 1, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2
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OFFSET
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0,3
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COMMENTS
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This sequence is equivalent to A002828 (least number of squares that add up to n) for sums and differences of squares. Here the possible forms include not only sums of squares, but also differences like x^2 - y^2 or x^2 + y^2 - z^2.
a(n) <= A002828(n) which is <= 4 (Lagrange's "Four Squares theorem"). In fact, a(n) <= 3: numbers of the form 4k, 4k+1 or 4k+3 are equal to the difference of two squares, therefore a(n) <= 2 in this case, and a(4k+2) <= 3 because 4k+2 = 4k+1+1^2. More precisely, a(4k) = 1 or 2; a(4k+1) = 1 or 2; a(4k+2) = 2 or 3; a(4k+3) = 2.
If A002828(n) = 4, a(n) = 2 (see A004215); if A002828(n) = 3, a(n) = 2 or 3: this shows that the form x^2 + y^2 - z^2 is never necessary to write an integer with the minimum number of squares; and of course, if A002828 = 1 or 2, a(n) = A002828.
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LINKS
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FORMULA
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Using the partition of the natural numbers into A000290 (square numbers), A000415 (sum of 2 nonzero squares), A263737 (difference but not sum of 2 squares) and A062316 (neither the sum or difference of 2 squares), the sequence is completely defined by: a(A000290(n)) = 1, a(A000415(n)) = a(A263737(n)) = 2, a(A062316(n)) = 3.
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EXAMPLE
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a(6) = 3 because 6 = 2^2 + 1^2 + 1^2 and 6 is not the sum or the difference of two squares; a(28) = 2 because 28 = 8^2 - 6^2.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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