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A246199 Odd half-Zumkeller numbers.

%I A246199 #26 Feb 13 2025 18:37:43
%S A246199 225,441,1225,2025,3969,5625,11025,18225,21609,27225,35721,38025,
%T A246199 50625,53361,65025,74529,81225,99225,119025,127449,140625,148225,
%U A246199 159201,164025,184041,189225,194481,207025,216225,233289,245025,275625,308025,314721,321489
%N A246199 Odd half-Zumkeller numbers.
%C A246199 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.
%D A246199 S. Clark et al., Zumkeller numbers, Mathematical Abundance Conference, April 2008.
%H A246199 Amiram Eldar, <a href="/A246199/b246199.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..503 from Chai Wah Wu)
%H A246199 K. P. S. Bhaskara Rao and Yuejian Peng, <a href="http://dx.doi.org/10.1016/j.jnt.2012.09.020">On Zumkeller Numbers</a>, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
%o A246199 (Python)
%o A246199 from sympy import divisors
%o A246199 import numpy as np
%o A246199 A246199 = []
%o A246199 for n in range(3, 10**5, 2):
%o A246199     d = divisors(n)
%o A246199     d.remove(n)
%o A246199     s, dmax = sum(d), max(d)
%o A246199     if not s % 2 and 2*dmax <= s:
%o A246199         d.remove(dmax)
%o A246199         s2, ld = int(s/2-dmax), len(d)
%o A246199         z = np.zeros((ld+1, s2+1), dtype=int)
%o A246199         for i in range(1, ld+1):
%o A246199             y = min(d[i-1], s2+1)
%o A246199             z[i, range(y)] = z[i-1, range(y)]
%o A246199             z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y)
%o A246199             if z[i, s2] == s2:
%o A246199                 A246199.append(n)
%o A246199                 break
%Y A246199 Odd numbers in A246198.
%Y A246199 Cf. A083207.
%K A246199 nonn
%O A246199 1,1
%A A246199 _Chai Wah Wu_, Aug 21 2014