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A352881
a(n) is the minimal number z having the largest number of solutions to the Diophantine equation 1/z = 1/x + 1/y such that 1 <= x <= y <= 10^n.
1
2, 12, 60, 840, 9240, 55440, 720720, 6126120, 116396280, 232792560, 5354228880, 26771144400, 465817912560, 4813451763120, 24067258815600, 144403552893600, 2671465728531600, 36510031623265200, 219060189739591200, 4709794079401210800, 18839176317604843200, 221360321731856907600
OFFSET
1,1
COMMENTS
Solving for z gives z = (x*y) / (x+y), so x*y == 0 (mod x+y).
All known terms are from A025487:
a(1) = 2 = 2;
a(2) = 12 = 2^2 * 3;
a(3) = 60 = 2^2 * 3 * 5;
a(4) = 840 = 2^3 * 3 * 5 * 7;
a(5) = 9240 = 2^3 * 3 * 5 * 7 * 11.
If a solution to the equation 1/z = 1/x + 1/y is found such that gcd(x,y,z) is a square, then x+y, x*y*z, and (x-y)^2 + (2*z)^2 are also squares.
For all solutions, x^2 + y^2 + z^2 is a square.
The sequence is indeed a subsequence of A025487, and likely of A126098 as well. - Max Alekseyev, Mar 01 2023
a(n) < 5*10^(n-1). - Max Alekseyev, Mar 01 2023
EXAMPLE
For n=1, we have the following, where r = (x*y) mod (x+y). (In the last four columns, each number marked by an asterisk is a square.)
.
r z x y x*y x+y x*y*z x^2+y^2+z^2
- - - - --- --- ----- -----------
0 1 2 2 4* 4* 4* 9* (solution)
2 1 2 4 8 6 8 21
4 1 2 6 12 8 12 41
6 1 2 8 16* 10 16* 69
3 1 3 3 9 6 9* 19
0 2 3 6 18* 9* 36* 49* (solution)
3 2 3 9 27 12 54 94
0 2 4 4 16* 8 32 36* (solution)
8 2 4 8 32 12 64* 84
5 2 5 5 25* 10 50 54
0 3 6 6 36* 12 108 81* (solution)
7 3 7 7 49* 14 147 107
0 4 8 8 64* 16* 256* 144* (solution)
9 4 9 9 81* 18 324* 178
.
z = 2 has the largest number of solutions, so a(1) = 2.
The number of solutions for the resulting z cannot exceed A018892(z).
PROG
(Python)
def a(n):
D = {}
for x in range(2, 10**n):
for y in range(x, 10**n):
z, r = divmod(x * y, x + y)
if r == 0:
if z in D:
D[z] += 1
else:
D[z] = 1
best_Z = 0
for z in D:
if D[z] > best_Z:
best_Z = D[z]
best_z = z
return best_z
KEYWORD
nonn
AUTHOR
Darío Clavijo, Apr 06 2022
EXTENSIONS
a(6) from Chai Wah Wu, Apr 10 2022
a(7)-a(22) from Max Alekseyev, Mar 01 2023
STATUS
approved