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

1,1

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.

Links

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

Example

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.

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);

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(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

Crossrefs

Cf. A083207, A066272, A192273.

Sequence in context: A115957 A362003 A255989 * A291319 A226168 A248430

Adjacent sequences: A192271 A192272 A192273 * A192275 A192276 A192277

Keyword

nonn

Author

Paolo P. Lava, Jun 28 2011

Extensions

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

Status

approved