login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 29 10:17 EDT 2022. Contains 354913 sequences. (Running on oeis4.)