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A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.

%I A345285 #34 Dec 24 2021 12:45:07
%S A345285 10,30,68,78,130,160,222,290,300,350,480,510,520,738,742,810,820,1010,
%T A345285 1088,1218,1248,1342,1530,1740,1752,1820,1830,2080,2210,2430,2560,
%U A345285 2590,2750,2758,3270,3390,3492,3552,3560,3570,4112,4290,4498,4640,4770,4800,4930,5508,5600,5850,6028,6250
%N A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.
%C A345285 Notation: s = side of the primary tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
%C A345285 Every term is of the form s = a*b * (a^2+b^2), with 1 <= a < b, and corresponding z = (a*b)^2 * (a^2+b^2) (A345286).
%C A345285 Every such primary square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.
%C A345285 If gcd(a, b) = 1, then primitive sides of square s = a*b * (a^2+b^2) are in A344333 that is a subsequence.
%C A345285 If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
%C A345285 If q is a term and integer r > 1, then q * r^4 is another term.
%C A345285 Every term is even.
%D A345285 Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
%H A345285 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A345285 a(1) = 10 and the primary square 10 X 10 can be tiled with A345286(1) = 20 small squares with side a = 1 and 20 large squares with side b = 2.
%e A345285       ___ ___ _ ___ ___ _
%e A345285      |   |   |_|   |   |_|
%e A345285      |___|___|_|___|___|_|
%e A345285      |   |   |_|   |   |_|     with 10 elementary 2 X 5 rectangles
%e A345285      |___|___|_|___|___|_|
%e A345285      |   |   |_|   |   |_|              ___ ___ _
%e A345285      |___|___|_|___|___|_|             |   |   |_|
%e A345285      |   |   |_|   |   |_|             |___|___|_|
%e A345285      |___|___|_|___|___|_|
%e A345285      |   |   |_|   |   |_|
%e A345285      |___|___|_|___|___|_|
%e A345285 a(6) = 160 is the first side of an primary square that is not primitive and it corresponds to (a,b) = (2,4); the square 160 X 160 can be tiled with A345286(6) = 1280 small squares with side a = 2 and 1280 large squares with side b = 4.
%Y A345285 Cf. A034262, A344330, A344333, A344334, A345286, A345287.
%Y A345285 Subsequence of A344331.
%K A345285 nonn
%O A345285 1,1
%A A345285 _Bernard Schott_, Jun 13 2021