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

1,1

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.

Links

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

Example

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.

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);

Prog

(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

Crossrefs

Cf. A083207, A066272, A192274.

Sequence in context: A175666 A299997 A299987 * A098748 A020692 A124187

Adjacent sequences: A192270 A192271 A192272 * A192274 A192275 A192276

Keyword

nonn

Author

Paolo P. Lava, Jun 28 2011

Extensions

Added missing terms from Chai Wah Wu, Aug 13 2014

Status

approved