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A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum. 43
6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A083206(a(n)) > 0; complement of A083210; subsequence of A023196; A083208(n) = A083206(a(n)).

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

A179529(a(n)) = 1. - Reinhard Zumkeller, Jul 19 2010

A118372 is a subsequence, the S-perfect numbers. - Reinhard Zumkeller, Oct 28 2010

All 205283 odd abundant numbers less than 10^8 that have even abundance 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

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.

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.

Farid Jokar, On the difference between Zumkeller numbers, arXiv:1902.02168 [math.NT], 2019.

Peter Luschny, Zumkeller Numbers.

Reinhard Zumkeller, Illustration of initial terms

EXAMPLE

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).

MAPLE

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:

A083207_list := upto -> select(is_A083207, [$1..upto]):

A083207_list(272); # Peter Luschny, Dec 14 2009, updated Aug 15 2014

MATHEMATICA

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 *)

PROG

(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

-- Reinhard Zumkeller, Apr 18 2013

(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

(Python3)

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

# Chai Wah Wu, Aug 13 2014

(Python)

from sympy import divisors

import numpy as np

A083207 = []

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:

................A083207.append(n)

................break

# Chai Wah Wu, Aug 19 2014

(Sage)

def is_Zumkeller(n):

    s = sigma(n)

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

    S = s // 2 - n

    R = filter(lambda m: m <= S, divisors(n))

    for c in Combinations(R):

        if sum(c) == S: return true

    return false

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

print A083207_list(272) # Peter Luschny, Sep 03 2018

CROSSREFS

Cf. A083209, A083211, A000203, A005101, A000396, A005835, A048055, A171641, A027750, A175592, A221054, A293453.

Sequence in context: A235268 A105455 A246198 * A304391 A145278 A094371

Adjacent sequences:  A083204 A083205 A083206 * A083208 A083209 A083210

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Apr 22 2003

EXTENSIONS

Name improved by T. D. Noe, Mar 31 2010

Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010

STATUS

approved

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Last modified March 21 03:23 EDT 2019. Contains 321359 sequences. (Running on oeis4.)