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%I #188 Jul 25 2024 11:06:23
%S 6,12,20,24,28,30,40,42,48,54,56,60,66,70,78,80,84,88,90,96,102,104,
%T 108,112,114,120,126,132,138,140,150,156,160,168,174,176,180,186,192,
%U 198,204,208,210,216,220,222,224,228,234,240,246,252,258,260,264,270,272
%N Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.
%C 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
%C 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 12. - _Ivan N. Ianakiev_, Jan 02 2024
%C All 205283 odd abundant numbers less than 10^8 that have even abundance are Zumkeller numbers. - _T. D. Noe_, Nov 14 2010
%C 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
%C Supersequence of A111592 (follows from Fact 3 in the Rao/Peng paper). - _Ivan N. Ianakiev_, Mar 20 2017
%C 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
%C 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
%C 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
%C 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
%C Conjecture: If d > 1, d|k and tau(d)*sigma(d) = k, then k is a Zumkeller number. - _Ivan N. Ianakiev_, Apr 24 2020
%D Marijo O. LeVan, Integer-perfect numbers, Journal of Natural Sciences and Mathematics, Vol. 27, No. 2 (1987), pp. 33-50.
%D Marijo O. LeVan, On the order of nu(n), Journal of Natural Sciences and Mathematics, Vol. 28, No. 1 (1988), pp. 165-173.
%D J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 1.10, pp. 53-54.
%H T. D. Noe, <a href="/A083207/b083207.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Basher, <a href="https://doi.org/10.1186/s42787-021-00121-y">k-Zumkeller labeling of super subdivision of some graphs</a>, J. Egyptian Math. Soc. (2021) Vol. 29, No. 12.
%H Hussein Behzadipour, <a href="https://arxiv.org/abs/1812.07233">Two-layered numbers</a>, arXiv:1812.07233 [math.NT], 2018.
%H K. P. S. Bhaskara Rao and Yuejian Peng, <a href="http://arxiv.org/abs/0912.0052">On Zumkeller Numbers</a>, arXiv:0912.0052 [math.NT], 2009.
%H K. P. S. Bhaskara Rao and Yuejian Peng, <a href="https://doi.org/10.1016/j.jnt.2012.09.020">On Zumkeller Numbers</a>, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
%H Bhabesh Das, <a href="https://doi.org/10.7546/nntdm.2024.30.2.436-442">On unitary Zumkeller numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 2, 436-442.
%H Farid Jokar, <a href="https://arxiv.org/abs/1902.02168">On the difference between Zumkeller numbers</a>, arXiv:1902.02168 [math.NT], 2019.
%H Farid Jokar, <a href="https://arxiv.org/abs/2003.11309">On k-layered numbers and some labeling related to k-layered numbers</a>, arXiv:2003.11309 [math.NT], 2020.
%H Farid Jokar, <a href="https://arxiv.org/abs/2207.09053">On k-layered numbers</a>, arXiv:2207.09053 [math.NT], 2022.
%H Peter Luschny, <a href="http://www.luschny.de/math/seq/ZumkellerNumbers.html">Zumkeller Numbers</a>.
%H Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, <a href="https://arxiv.org/abs/2008.11096">Some properties of Zumkeller numbers and k-layered numbers</a>, arXiv:2008.11096 [math.NT], 2020.
%H Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, <a href="https://doi.org/10.1016/j.jnt.2020.05.003">Some properties of Zumkeller numbers and k-layered numbers</a>, Journal of Number Theory (2020).
%H Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, <a href="https://arxiv.org/abs/2310.14149">Some Results on Zumkeller Numbers</a>, arXiv:2310.14149 [math.NT], 2023.
%H Reinhard Zumkeller, <a href="/A083206/a083206.txt">Illustration of initial terms</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F A083206(a(n)) > 0.
%F A083208(n) = A083206(a(n)).
%F A179529(a(n)) = 1. - _Reinhard Zumkeller_, Jul 19 2010
%e 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).
%p with(numtheory): with(combstruct):
%p is_A083207 := proc(n) local S, R, Found, Comb, a, s; s := sigma(n);
%p if not(modp(s, 2) = 0 and n * 2 <= s) then RETURN(false) fi;
%p S := s / 2 - n; R := select(m -> m <= S, divisors(n)); Found := false;
%p Comb := iterstructs(Combination(R)):
%p while not finished(Comb) and not Found do
%p Found := add(a, a = nextstruct(Comb)) = S
%p od; Found end:
%p A083207_list := upto -> select(is_A083207, [$1..upto]):
%p A083207_list(272); # _Peter Luschny_, Dec 14 2009, updated Aug 15 2014
%t 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 *)
%t 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 *)
%o (Haskell)
%o a083207 n = a083207_list !! (n-1)
%o a083207_list = filter (z 0 0 . a027750_row) $ [1..] where
%o z u v [] = u == v
%o z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
%o -- _Reinhard Zumkeller_, Apr 18 2013
%o (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]
%o 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
%o (Python)
%o from sympy import divisors
%o from sympy.combinatorics.subsets import Subset
%o for n in range(1,10**3):
%o d = divisors(n)
%o s = sum(d)
%o if not s % 2 and max(d) <= s/2:
%o for x in range(1,2**len(d)):
%o if sum(Subset.unrank_binary(x,d).subset) == s/2:
%o print(n,end=', ')
%o break
%o # _Chai Wah Wu_, Aug 13 2014
%o (Python)
%o from sympy import divisors
%o import numpy as np
%o A083207 = []
%o for n in range(2,10**3):
%o d = divisors(n)
%o s = sum(d)
%o if not s % 2 and 2*n <= s:
%o d.remove(n)
%o s2, ld = int(s/2-n), len(d)
%o z = np.zeros((ld+1,s2+1),dtype=int)
%o for i in range(1,ld+1):
%o y = min(d[i-1],s2+1)
%o z[i,range(y)] = z[i-1,range(y)]
%o z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
%o if z[i,s2] == s2:
%o A083207.append(n)
%o break
%o # _Chai Wah Wu_, Aug 19 2014
%o (Sage)
%o def is_Zumkeller(n):
%o s = sigma(n)
%o if not (2.divides(s) and n*2 <= s): return False
%o S = s // 2 - n
%o R = (m for m in divisors(n) if m <= S)
%o return any(sum(c) == S for c in Combinations(R))
%o A083207_list = lambda lim: [n for n in (1..lim) if is_Zumkeller(n)]
%o print(A083207_list(272)) # _Peter Luschny_, Sep 03 2018
%Y Complement of A083210.
%Y Subsequence of A023196 and of A028983.
%Y A118372 is a subsequence.
%Y Cf. A204830, A204831 (equal sums of 3 or 4 disjoint subsets).
%Y Cf. A083206, A083208, A083209, A083211, A000203, A005101, A000396, A005835, A048055, A171641, A027750, A175592, A221054, A293453.
%K nonn,nice
%O 1,1
%A _Reinhard Zumkeller_, Apr 22 2003
%E Name improved by _T. D. Noe_, Mar 31 2010
%E Name "Zumkeller numbers" added by _N. J. A. Sloane_, Jul 08 2010
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