A192273 Zumkeller numbers using anti-divisors (or anti-Zumkeller numbers).

7, 10, 17, 22, 23, 31, 32, 33, 35, 37, 38, 39, 42, 45, 49, 50, 52, 53, 55, 58, 63, 67, 68, 70, 72, 73, 77, 78, 82, 83, 87, 88, 93, 94, 95, 98, 103, 105, 115, 116, 117, 123, 126, 127, 128, 130, 137, 142, 143, 148, 149, 157, 158, 160, 162, 163, 165, 171, 175

Offset: 1

Author

Paolo P. Lava, Jun 28 2011

Keywords

• nonn

Comments

Numbers n whose anti-divisors can be partitioned into two disjoint sets whose sums are both sigma*(n)/2, where sigma*(n) is the sum of the anti-divisors of n.

Examples

			7 -> anti-divisors: 2,3,5; sigma*(7)/2 = 5; 2+3 = 5.
77-> anti-divisors: 2,3,5,9,14,17,22,31,51; sigma*(77)/2=77; 2+3+5+14+22+31=9+17+51=77.
143->anti-divisors: 2,3,5,7,15,19,22,26,41,57,95;  sigma*(143)/2=146; 2+5+15+19+22+26+57=3+7+41+95=146.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A083207, A066272, A192274.

Programs

Maple

with(combstruct);
P:=proc(i)
local S,R,Stop,Comb,a,b,c,d,k,m,n,s;
for n from 3 to i do
  a:={};
  for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi; od;
  b:=nops(a); c:=op(a); s:=0;
  if b>1 then for k from 1 to b do s:=s+c[k]; od;
  else s:=c;
  fi;
  if (modp(s,2)=0 and 2*n<=s) then
      S:=1/2*s-n; R:=select(m->m<=S,[c]); Stop:=false; Comb:=iterstructs(Combination(R));
     while not (finished(Comb) or Stop) do Stop:=add(d,d=nextstruct(Comb))=S; od;
     if Stop then print(n); fi;
  fi;
od;
end:
P(3000);

Python

from sympy import divisors
from sympy.combinatorics.subsets import Subset
def antidivisors(n):
    return [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \
           [d for d in divisors(2*n-1) if n > d >=2 and n % d] + \
           [d for d in divisors(2*n+1) if n > d >=2 and n % d]
for n in range(3,10**3):
    d = antidivisors(n)
    s = sum(d)
    if not s % 2 and max(d) <= s//2:
        for x in range(1,2**len(d)):
            if sum(Subset.unrank_binary(x,d).subset) == s//2:
                print(n,end=', ')
                break
# Chai Wah Wu, Aug 13 2014

Python

from sympy import divisors
import numpy as np
A192273 = []
for n in range(3,10**3):
    d = [2*x for x in divisors(n) if n > 2*x and n % (2*x)] + \
        [x for x in divisors(2*n-1) if n > x >=2 and n % x] + \
        [x for x in divisors(2*n+1) if n > x >=2 and n % x]
    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:
                A192273.append(n)
                break
# Chai Wah Wu, Aug 19 2014

Extensions

Added missing terms from Chai Wah Wu, Aug 13 2014