OFFSET
1,2
COMMENTS
Possner (see also Knuth) asks for solutions when n=2. There is at least one solution for all positive n: (x,y,z) = (n+1, n^2+2n, n^3+3n^2+n). All solutions appear to be in the polytope n < x <= 2n^2 + n, x < y <= 2n^3 + 2n^2 - n, y < z <= n^5 + 2n^4 + 2n^3 + n^2 - n. Many solutions, especially for prime n, are such that n divides x, y and z. See A094595.
Following the linked solution by Silvia Fernández, it can be shown that all solutions satisfy n < x <= 3n^2 - n, x < y <= 2nx - n, and y < z <= xy - n. Contrary to the above comment, there are solutions satisfying x > 2n^2 + n. The first example is given by (x,y,z) = (434,630,826) when n = 14. - Robin Visser, Dec 18 2023
LINKS
M. F. Possner, Problem 11021, Amer. Math. Monthly, 110 (2003), p. 542.
Donald Knuth, Silvia Fernández and Gerry Myerson, A Modular Triple: 11021, Amer. Math. Monthly, 112 (2005), p. 279.
EXAMPLE
a(2) = 6 because there are 6 solutions: (x,y,z) = (3, 8, 22), (3, 10, 14), (4, 5, 18), (4, 6, 11), (6, 14, 82) and (6, 22, 26).
PROG
(Sage)
def a(n):
ans = 0
for x in range(n+1, 3*n^2-n+1):
for y in range(x+1, 2*n*x-n+1):
for z in Integer(x*y-n).divisors():
if (z > y) and ((x*y)%z)==n and ((y*z)%x)==n and ((z*x)%y)==n:
ans += 1
return ans # Robin Visser, Dec 12 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, May 06 2004, revised May 13 2004
EXTENSIONS
Corrected and a(33)-a(46) added by Robin Visser, Dec 18 2023
a(47)-a(61) from Robin Visser, Mar 18 2024
STATUS
approved