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 A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum. 95
 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 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. - Ivan N. Ianakiev, Apr 24 2020 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. 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). 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, 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 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 = (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 CROSSREFS Cf. A083209, A083211, A000203, A005101, A000396, A005835, A048055, A171641, A027750, A175592, A221054, A293453. Sequence in context: A105455 A345919 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 September 28 20:34 EDT 2022. Contains 357081 sequences. (Running on oeis4.)