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A263765 Minimum number of squares necessary to write n as a sum or difference of squares. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 0..10000

FORMULA

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 define by: a(A000290(n)) = 1, a(A000415(n)) = a(A263737(n)) = 2, a(A062316(n)) = 3.

EXAMPLE

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.

CROSSREFS

Cf. A002828, A000290, A000415, A062316, A263715, A263737.

Sequence in context: A309287 A056173 A216817 * A270073 A027348 A238325

Adjacent sequences:  A263762 A263763 A263764 * A263766 A263767 A263768

KEYWORD

nonn

AUTHOR

Jean-Christophe Hervé, Oct 25 2015

STATUS

approved

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Last modified August 25 16:50 EDT 2019. Contains 326324 sequences. (Running on oeis4.)