The 229026 Zumkeller numbers less than 10^6 have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than 10^6; they are exactly the odd abundant numbers that have even abundance, A174865. - T. D. Noe, Mar 31 2010

For k >= 0, numbers of the form 18k + 6 and 18k + 12 are terms (see Remark 2.3. in Somu et al., 2023). Corollary: The maximum difference between any two consecutive terms is at most 12. - Ivan N. Ianakiev, Jan 02 2024

All 205283 odd abundant numbers less than 10^8 that have even abundance (see A174865) are Zumkeller numbers. - T. D. Noe, Nov 14 2010

Except for 1 and 2, all primorials (A002110) are Zumkeller numbers (follows from Fact 6 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 23 2016

Supersequence of A111592 (follows from Fact 3 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 20 2017

Conjecture: Any 4 consecutive terms include at least one number k such that sigma(k)/2 is also a Zumkeller number (verified for the first 10^5 Zumkeller numbers). - Ivan N. Ianakiev, Apr 03 2017

LeVan studied these numbers using the equivalent definition of numbers n such that n = Sum_{d|n, d<n} alpha(d)*d, where alpha(d) is either 1 or -1, and named them "integer-perfect numbers". She also named the primitive Zumkeller numbers (A180332) "minimal integer-perfect numbers". - Amiram Eldar, Dec 20 2018

The numbers 3 * 2^k for k > 0 are all Zumkeller numbers: half of one such partition is {3*2^k, 3*2^(k-2), ...}, replacing 3 with 2 if it appears. With this and the lemma that the product of a Zumkeller number and a number coprime to it is again a Zumkeller number (see A179527), we have that all numbers divisible by 6 but not 9 (or numbers congruent to 6 or 12 modulo 18) are Zumkeller numbers, proving that the difference between consecutive Zumkeller numbers is at most 12. - Charlie Neder, Jan 15 2019

Improvements on the previous comment: 1) For every integer q > 0, every odd integer r > 0 and every integer s > 0 relatively prime to 6, the integer 2^q*3^r*s is a Zumkeller number, and therefore 2) there exist Zumkeller numbers divisible by 9 (such as 54, 90, 108, 126, etc.). - Ivan N. Ianakiev, Jan 16 2020

Conjecture: If d > 1, d|k and tau(d)*sigma(d) = k, then k is a Zumkeller number (cf. A331668). - Ivan N. Ianakiev, Apr 24 2020

This sequence contains A378541, the intersection of the practical numbers (A005153) with numbers with even sum of divisors (A028983). - David A. Corneth, Nov 03 2024

Sequence gives the positions of even terms in A119347, and correspondingly, of odd terms in A308605. - Antti Karttunen, Nov 29 2024

If s = sigma(m) is odd and p > s then m*p is not in the sequence. - David A. Corneth, Dec 07 2024

Marijo O. LeVan, Integer-perfect numbers, Journal of Natural Sciences and Mathematics, Vol. 27, No. 2 (1987), pp. 33-50.

Marijo O. LeVan, On the order of nu(n), Journal of Natural Sciences and Mathematics, Vol. 28, No. 1 (1988), pp. 165-173.

J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 1.10, pp. 53-54.

K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, arXiv:0912.0052 [math.NT], 2009.

K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Given n = 48, we can partition the divisors thus: 1 + 3 + 4 + 6 + 8 + 16 + 24 = 2 + 12 + 48, therefore 48 is a term (A083206(48) = 5).

30 is in the sequence. sigma(30) = 72. So we look for distinct divisors of 30 that sum to 72/2 = 36. That set or its complement contains 30. The other divisors in that set containing 30 sum to 36 - 30 = 6. So we look for some distinct proper divisors of 30 that sum to 6. That is from the divisors of {1, 2, 3, 5, 6, 10, 15}. It turns out that both 1+2+3 and 6 satisfy this condition. So 36 is in the sequence.

25 is not in the sequence as sigma(25) = 31 which is odd so the sum of two equal integers cannot be the sum of divisors of 25.

33 is not in the sequence as sigma(33) = 48 < 2*33. So is impossible to have a partition of the set of divisors into two disjoint set the sum of each of them sums to 48/2 = 24 as one of them contains 33 > 24 and any other divisors are nonnegative. (End)

with(numtheory): with(combstruct):

is_A083207 := proc(n) local S, R, Found, Comb, a, s; s := sigma(n);

if not(modp(s, 2) = 0 and n * 2 <= s) then return false fi;

S := s / 2 - n; R := select(m -> m <= S, divisors(n)); Found := false;

Comb := iterstructs(Combination(R)):

while not finished(Comb) and not Found do

Found := add(a, a = nextstruct(Comb)) = S

od; Found end:

ZumkellerQ[n_] := Module[{d=Divisors[n], t, ds, x}, ds = Plus@@d; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[1000], ZumkellerQ] (* T. D. Noe, Mar 31 2010 *)

znQ[n_]:=Length[Select[{#, Complement[Divisors[n], #]}&/@Most[Rest[ Subsets[ Divisors[ n]]]], Total[#[[1]]]==Total[#[[2]]]&]]>0; Select[Range[300], znQ] (* Harvey P. Dale, Dec 26 2022 *)

(Haskell)

a083207 n = a083207_list !! (n-1)

a083207_list = filter (z 0 0 . a027750_row) $ [1..] where

z u v [] = u == v

z u v (p:ps) = z (u + p) v ps || z u (v + p) ps

(PARI) part(n, v)=if(n<1, return(n==0)); forstep(i=#v, 2, -1, if(part(n-v[i], v[1..i-1]), return(1))); n==v[1]

is(n)=my(d=divisors(n), s=sum(i=1, #d, d[i])); s%2==0 && part(s/2-n, d[1..#d-1]) \\ Charles R Greathouse IV, Mar 09 2014

(PARI) \\ See Corneth link

(Python)

from sympy import divisors

from sympy.combinatorics.subsets import Subset

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:

print(n, end=', ')

break

(Python)

from sympy import divisors

import numpy as np

for n in range(2, 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:

break

(SageMath)

def is_Zumkeller(n):

s = sigma(n)

if not (2.divides(s) and n*2 <= s): return False

S = s // 2 - n

R = (m for m in divisors(n) if m <= S)

return any(sum(c) == S for c in Combinations(R))

A083207_list = lambda lim: [n for n in (1..lim) if is_Zumkeller(n)]

Positions of even terms in A119347, of odd terms in A308605.

Conjectured subsequences: A007691, A331668 (after their initial 1's), A351548 (apart from 0-terms).

Cf. A174865 (Odd abundant numbers whose abundance is even).

Cf. A204830, A204831 (equal sums of 3 or 4 disjoint subsets).