A344333 - OEIS

%I #16 Dec 23 2021 10:03:42

%S 10,30,68,78,130,222,290,300,350,510,520,738,742,820,1010,1218,1342,

%T 1530,1740,1752,1820,1830,2210,2590,2750,2758,3270,3390,3492,3560,

%U 3570,4112,4290,4498,4770,4930,5850,6028,6328,6870,6878,6942,8020,8030,8190,8610,9282,9620,9962

%N Primitive side of squares of type 1 (A344331) that are tiled with squares of two different sizes so that the number of large or small squares is the same.

%C Some notations: s = side of the tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.

%C Every term is of the form s = a*b * (a^2+b^2) with gcd(a, b) = 1, then corresponding z = (a*b)^2 * (a^2+b^2) (see A344334).

%C Every primitive 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 If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.

%C Every term is even.

%D Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

%e Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.

%e       ___ ___ _ ___ ___ _

%e      |   |   |_|   |   |_|

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%e      |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles

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%e      |   |   |_|   |   |_|              ___ ___ _

%e      |___|___|_|___|___|_|             |   |   |_|

%e      |   |   |_|   |   |_|             |___|___|_|

%e      |___|___|_|___|___|_|

%e      |   |   |_|   |   |_|

%e      |___|___|_|___|___|_|

%o (PARI) isok(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->if (gcd(x,y)==1, x*y*(x^2+y^2), 0), [1..m]), s); } \\ _Michel Marcus_, Dec 22 2021

%Y Cf. A034262, A344330, A344331, A344332, A344334.

%K nonn

%O 1,1

%A _Bernard Schott_, Jun 01 2021