OFFSET
1,3
COMMENTS
These squares with side = A344330(n) can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.
Notation: 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.
REFERENCES
Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
LINKS
FORMULA
EXAMPLE
-> A344330(1) = A344331(1) = 10 and there is no k_2 such that A344330(1) = A344332(k_2) = 10, then a(1) = A345287(1) = 1 (example below of type 1):
Primitive square 10 X 10 corresponding to a(1) = 1 with
a = 1, b = 2, s = 10, z = 20:
___ ___ _ ___ ___ _
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
-> A344330(2) = A344332(1) = 15 and there is no k_1 such that A344330(2) = A344331(k_1) = 15, then a(2) = A346264(1) = 1 (example below of type 2):
Primitive square 15 X 15 corresponding to a(2) = 1 with
a = 3, b = 4, c = 5, s = 15, z = 9:
________ ________ ________ ______
| | | | |
| | | | |
| | | |______|
|_______ |________|________| |
| | | | |
| | | |______|
| | | | |
|________|________|________| |
| | | |______|
| | | | |
| | | | |
|_____ __|___ ____|_ ______|______|
| | | | | |
| | | | | |
|_____|______|______|______|______|
PROG
isok3(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); } \\ A344333
sd7(x) = sumdiv(x, d, if (isok3(d), numdiv(x/d))); \\ A345287
isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k); \\ A005917
sd4(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d))); \\ A346264
lista(nn) = {for (n=1, nn, my(b1 = isok1(n), b2 = isok2(n)); if (b1 || b2, my(x = 0); if (b1, x += sd7(n)); if (b2, x += sd4(n)); print1(x, ", "); ); ); } \\ Michel Marcus, Dec 23 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Aug 11 2021
EXTENSIONS
a(19),a(59),a(86),a(87) corrected by Bernard Schott, Dec 23 2021
STATUS
approved