|
%I #37 Aug 26 2020 03:06:43
%S 6,12,20,24,28,30,40,42,48,54,56,60,66,70,78,80,84,88,90,96,102,104,
%T 108,112,114,120,126,132,138,140,150,156,160,168,174,176,180,186,192,
%U 198,204,208,210,216,220,222,224,225,228,234,240,246,252,258,260,264
%N Half-Zumkeller numbers: numbers n whose proper positive divisors can be partitioned into two disjoint sets whose sums are equal.
%C 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.
%D S. Clark et al., Zumkeller numbers, Mathematical Abundance Conference, April 2008.
%H Robert Israel, <a href="/A246198/b246198.txt">Table of n, a(n) for n = 1..9188</a> (n=1..1309 from Chai Wah Wu)
%H 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.
%H Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, <a href="https://arxiv.org/abs/2008.11096">Some properties of Zumkeller numbers and k-layered numbers</a>, arXiv:2008.11096 [math.NT], 2020.
%e Proper divisors of 225 are 1, 3, 5, 9, 15, 25, 45, 75 and 1+3+15+25+45=5+9+75.
%p filter:= proc(n) local L,s,t,nL,B,j,k;
%p L:= numtheory:-divisors(n) minus {n};
%p s:= convert(L,`+`);
%p if s::odd then return false fi;
%p t:= s/2;
%p nL:= nops(L);
%p B:= Array(0..t,1..nL);
%p B[0,1]:= 1;
%p B[L[1],1]:= 1;
%p for j from 2 to nL do
%p B[..,j]:= B[..,j-1];
%p for k from L[j] to t do
%p B[k,j]:= B[k,j] + B[k-L[j],j-1]
%p od:
%p if B[t,j] > 0 then return true fi;
%p od:
%p false
%p end:
%p select(filter, [$2..300]); # _Robert Israel_, Aug 19 2014
%t filterQ[n_] := Module[{L, s, t, nL, B, j, k},
%t L = Most[Divisors[n]];
%t s = Total[L];
%t If[OddQ[s], Return[False]];
%t t = s/2;
%t nL = Length[L];
%t B[_, _] = 0;
%t B[0, 1] = 1;
%t B[L[[1]], 1] = 1;
%t For[j = 2, j <= nL, j++,
%t Do[B[k, j] = B[k, j-1], {k, 0, t}];
%t For[k = L[[j]], k <= t, k++,
%t B[k, j] = B[k, j] + B[k-L[[j]], j-1]
%t ];
%t If[ B[t, j] > 0, Return[True]];
%t ];
%t False
%t ];
%t Select[Range[2, 300], filterQ] (* _Jean-François Alcover_, Mar 04 2019, after _Robert Israel_ *)
%t 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 *)
%o (Python)
%o from sympy.combinatorics.subsets import Subset
%o from sympy import divisors
%o A246198 = []
%o for n in range(2,10**3):
%o ....d = divisors(n)
%o ....d.remove(n)
%o ....s, dmax = sum(d), max(d)
%o ....if not s % 2 and 2*dmax <= s:
%o ........d.remove(dmax)
%o ........s2 = s/2-dmax
%o ........for x in range(2**len(d)):
%o ............if sum(Subset.unrank_binary(x,d).subset) == s2:
%o ................A246198.append(n)
%o ................break
%o (Python)
%o from sympy import divisors
%o import numpy as np
%o A246198 = []
%o for n in range(2,10**3):
%o ....d = divisors(n)
%o ....d.remove(n)
%o ....s, dmax = sum(d), max(d)
%o ....if not s % 2 and 2*dmax <= s:
%o ........d.remove(dmax)
%o ........s2, ld = int(s/2-dmax), len(d)
%o ........z = np.zeros((ld+1,s2+1),dtype=int)
%o ........for i in range(1,ld+1):
%o ............y = min(d[i-1],s2+1)
%o ............z[i,range(y)] = z[i-1,range(y)]
%o ............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 ............if z[i,s2] == s2:
%o ................A246198.append(n)
%o ................break
%o # _Chai Wah Wu_, Aug 19 2014
%Y Cf. A083207.
%K nonn
%O 1,1
%A _Chai Wah Wu_, Aug 18 2014
|
