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.

10, 30, 68, 78, 130, 160, 222, 290, 300, 350, 480, 510, 520, 738, 742, 810, 820, 1010, 1088, 1218, 1248, 1342, 1530, 1740, 1752, 1820, 1830, 2080, 2210, 2430, 2560, 2590, 2750, 2758, 3270, 3390, 3492, 3552, 3560, 3570, 4112, 4290, 4498, 4640, 4770, 4800, 4930, 5508, 5600, 5850, 6028, 6250

Offset

1,1

Comments

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.

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).

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.

If gcd(a, b) = 1, then primitive sides of square s = a*b * (a^2+b^2) are in A344333 that is a subsequence.

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

If q is a term and integer r > 1, then q * r^4 is another term.

Every term is even.

References

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

Links

Table of n, a(n) for n=1..52.

Index to sequences related to Olympiads.

Example

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.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|     with 10 elementary 2 X 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
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.

Crossrefs

Cf. A034262, A344330, A344333, A344334, A345286, A345287.

Subsequence of A344331.

Sequence in context: A255601 A104044 A124080 * A344333 A034127 A229466

Adjacent sequences: A345282 A345283 A345284 * A345286 A345287 A345288

Keyword

nonn

Author

Bernard Schott, Jun 13 2021

Status

approved