A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.
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
Offset: 1
Examples
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).
From _David A. Corneth_, Dec 04 2024: (Start)
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)
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
- M. Basher, k-Zumkeller labeling of super subdivision of some graphs, J. Egyptian Math. Soc. (2021) Vol. 29, No. 12.
- 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.
- David A. Corneth, PARI programs (Use function "is" for A179527)
- Bhabesh Das, On unitary Zumkeller numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 2, 436-442.
- Farid Jokar, On the difference between Zumkeller numbers, arXiv:1902.02168 [math.NT], 2019.
- Farid Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
- Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.
- Peter Luschny, Zumkeller Numbers.
- Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, arXiv:2008.11096 [math.NT], 2020.
- Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, Journal of Number Theory (2020).
- Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some Results on Zumkeller Numbers, arXiv:2310.14149 [math.NT], 2023.
- Reinhard Zumkeller, Illustration of initial terms
- Index entries for sequences where any odd perfect numbers must occur
Programs
-
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
-
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 *) 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 *) -
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 # 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 = (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)] print(A083207_list(272)) # Peter Luschny, Sep 03 2018
Formula
Extensions
Name improved by T. D. Noe, Mar 31 2010
Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010
Comments