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A192273 Zumkeller numbers using anti-divisors (or anti-Zumkeller numbers) 2
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 (list; graph; refs; listen; history; text; internal format)
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

(Python3)

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

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Last modified October 14 12:02 EDT 2019. Contains 328004 sequences. (Running on oeis4.)