A246198 Half-Zumkeller numbers: numbers n whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal.

6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 225, 228, 234, 240, 246, 252, 258, 260, 264

Offset: 1

Author

Chai Wah Wu, Aug 18 2014

Keywords

• nonn

Comments

All even half-Zumkeller numbers are in A083207, i.e. they are Zumkeller numbers (see Clark et al. 2008). The first 47 terms coincide with A083207. 225 is the first number in the sequence that is not a Zumkeller number.

Examples

			Proper divisors of 225 are 1, 3, 5, 9, 15, 25, 45, 75 and 1+3+15+25+45=5+9+75.
		

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..9188 (n=1..1309 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.
• Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, arXiv:2008.11096 [math.NT], 2020.

Crossrefs

Cf. A083207.

Programs

Maple

filter:= proc(n) local L,s,t,nL,B,j,k;
   L:= numtheory:-divisors(n) minus {n};
   s:= convert(L,`+`);
   if s::odd then return false fi;
   t:= s/2;
   nL:= nops(L);
   B:= Array(0..t,1..nL);
   B[0,1]:= 1;
   B[L[1],1]:= 1;
   for j from 2 to nL do
      B[..,j]:= B[..,j-1];
      for k from L[j] to t do
         B[k,j]:= B[k,j] + B[k-L[j],j-1]
      od:
      if B[t,j] > 0 then return true fi;
   od:
   false
end:
select(filter, [$2..300]); # Robert Israel, Aug 19 2014

Mathematica

filterQ[n_] := Module[{L, s, t, nL, B, j, k},
  L = Most[Divisors[n]];
  s = Total[L];
  If[OddQ[s], Return[False]];
  t = s/2;
  nL = Length[L];
  B[, ] = 0;
  B[0, 1] = 1;
  B[L[[1]], 1] = 1;
  For[j = 2, j <= nL, j++,
    Do[B[k, j] = B[k, j-1], {k, 0, t}];
    For[k = L[[j]], k <= t, k++,
      B[k, j] = B[k, j] + B[k-L[[j]], j-1]
    ];
    If[ B[t, j] > 0, Return[True]];
  ];
  False
];
Select[Range[2, 300], filterQ] (* Jean-François Alcover, Mar 04 2019, after Robert Israel *)
hzQ[n_] := Module[{d = Most @ Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[2, 1000], hzQ] (* Amiram Eldar, May 03 2020 *)

Python

from sympy.combinatorics.subsets import Subset
from sympy import divisors
A246198 = []
for n in range(2,10**3):
    d = divisors(n)
    d.remove(n)
    s, dmax = sum(d), max(d)
    if not s % 2 and 2*dmax <= s:
        d.remove(dmax)
        s2 = s/2-dmax
        for x in range(2**len(d)):
            if sum(Subset.unrank_binary(x,d).subset) == s2:
                A246198.append(n)
                break

Python

from sympy import divisors
import numpy as np
A246198 = []
for n in range(2, 10**3):
    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:
                A246198.append(n)
                break
# Chai Wah Wu, Aug 19 2014