OFFSET
1,1
COMMENTS
There are no subsets for m = 4 * k + 6 and just one subset for prime numbers m.
REFERENCES
Wilfried Haag, Die Wurzel. Problem 2020 - 14. (March/April 2020: www.wurzel.org)
FORMULA
For m = 6 * k + 3, you can always find two subsets of {v, ...,v+m-1} of length 3 * k + 2 with v = (3 * k + 1)^2 and length 3 * k + 3 with v = 3*k^2-1 elements.
EXAMPLE
m = 9, v=16: 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24.
m = 9, v=2: 2 + 3 + 4 + 5 + 6 + 7 = 8 + 9 + 10 = 27.
PROG
(Python)
for m in range(3, 1000):
anz = 0
for i in range(m // 2 + 1, m):
l = (2 * i * i - 2 * i - m * (m - 1)) / (2 * (m - 2 * i))
if l - int(l) == 0 and l >= 0:
anz = anz + 1
if anz > 1:
print(m)
CROSSREFS
KEYWORD
nonn
AUTHOR
Reiner Moewald, Mar 14 2020
STATUS
approved