A246199 Odd half-Zumkeller numbers.

225, 441, 1225, 2025, 3969, 5625, 11025, 18225, 21609, 27225, 35721, 38025, 50625, 53361, 65025, 74529, 81225, 99225, 119025, 127449, 140625, 148225, 159201, 164025, 184041, 189225, 194481, 207025, 216225, 233289, 245025, 275625, 308025, 314721, 321489

Offset: 1

Author

Chai Wah Wu, Aug 21 2014

Keywords

• nonn

Comments

Zumkeller numbers are numbers whose positive divisors can be partitioned into two disjoint sets whose sums are equal (A083207). Half-Zumkeller numbers are numbers whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal (A246198). All numbers in the sequence are not Zumkeller numbers. This is easily seen as the sum of proper divisors is even to be half-Zumkeller, and therefore the sum of the divisors must be odd and thus not Zumkeller.

References

• S. Clark et al., Zumkeller numbers, Mathematical Abundance Conference, April 2008.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..503 from Chai Wah Wu)
• K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Crossrefs

Odd numbers in A246198.

Cf. A083207.

Programs

Python

from sympy import divisors
import numpy as np
A246199 = []
for n in range(3, 10**5, 2):
    d = divisors(n)
    d.remove(n)
    s, dmax = sum(d), max(d)
    if not s % 2 and 2*dmax <= s:
        d.remove(dmax)
        s2, ld = int(s/2-dmax), len(d)
        z = np.zeros((ld+1, s2+1), dtype=int)
        for i in range(1, ld+1):
            y = min(d[i-1], s2+1)
            z[i, range(y)] = z[i-1, range(y)]
            z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y)
            if z[i, s2] == s2:
                A246199.append(n)
                break