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A083207 Zumkeller numbers: numbers n whose divisors can be partitioned into two disjoint sets whose sums are both sigma(n)/2. 34
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

LINKS

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

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

Peter Luschny, Zumkeller Numbers.

Reinhard Zumkeller, Illustration of initial terms

EXAMPLE

n=48: 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(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

CROSSREFS

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

Sequence in context: A235268 A105455 A246198 * A145278 A094371 A189793

Adjacent sequences:  A083204 A083205 A083206 * A083208 A083209 A083210

KEYWORD

nonn

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 April 27 06:52 EDT 2017. Contains 285508 sequences.