A192274 Numbers which are both Zumkeller numbers and anti-Zumkeller numbers.

42, 70, 78, 88, 126, 160, 176, 228, 234, 258, 270, 280, 308, 342, 350, 368, 378, 380, 390, 396, 402, 438, 448, 462, 468, 490, 500, 522, 532, 540, 552, 558, 560, 572, 580, 588, 608, 618, 620, 630, 644, 650, 690, 702, 732, 756, 770, 780, 798, 812, 822, 852, 858

Offset: 1

Author

Paolo P. Lava, Jun 28 2011

Keywords

• nonn

Comments

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

Examples

			270-> divisors: 1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270; sigma(270)/2=360; 1+2+3+5+6+9+10+15+18+27+30+45+54+135=90+270=360.
270-> anti-divisors: 4,7,11,12,20,36,49,60,77,108,180; sigma*(270)/2=282; 4+7+11+20+60+180=12+36+49+77+108=282.
		

Links

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

Crossrefs

Cf. A083207, A066272, A192273.

Programs

Maple

with(combstruct);
with(numtheory);
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
        s:=sigma(n);
        if (modp(s,2)=0 and 2*n<=s) then
          S:=1/2*s-n; R:=select(m->m<=S,divisors(n)); 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;
     fi;
  fi;
od;
end:
P(10000);

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(1, 10**3):
    d = divisors(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:
                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
                break
# Chai Wah Wu, Aug 14 2014

Python

from sympy import divisors
import numpy as np
A192274 = []
for n in range(3,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s:
        d.remove(n)
        s2, ld = int(s/2-n), 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:
                d2 = [2*x for x in d 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(d2), max(d2)
                if not s % 2 and 2*dmax <= s:
                    d2.remove(dmax)
                    s2, ld = int(s/2-dmax), len(d2)
                    z = np.zeros((ld+1,s2+1),dtype=int)
                    for i in range(1,ld+1):
                        y = min(d2[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:
                            A192274.append(n)
                        break
                break
# Chai Wah Wu, Aug 19 2014

Extensions

Corrected entries and comment by Chai Wah Wu, Aug 13 2014