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A263765 Minimum number of squares necessary to write n as a sum or difference of squares.

%I A263765 #13 Jun 27 2023 15:08:18
%S A263765 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,
%T A263765 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,
%U A263765 2,2,3,2,2,2,2,2,2
%N A263765 Minimum number of squares necessary to write n as a sum or difference of squares.
%C A263765 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.
%C A263765 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.
%C A263765 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.
%H A263765 Jean-Christophe Hervé, <a href="/A263765/b263765.txt">Table of n, a(n) for n = 0..10000</a>
%F A263765 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.
%e A263765 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.
%Y A263765 Cf. A002828, A000290, A000415, A062316, A263715, A263737.
%K A263765 nonn
%O A263765 0,3
%A A263765 _Jean-Christophe Hervé_, Oct 25 2015