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

1,1

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.

Prog

(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

Crossrefs

Odd numbers in A246198.

Cf. A083207.

Sequence in context: A373087 A216419 A390955 * A147276 A219022 A193003

Adjacent sequences: A246196 A246197 A246198 * A246200 A246201 A246202

Keyword

nonn,changed

Author

Chai Wah Wu, Aug 21 2014

Status

approved