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A083207
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Zumkeller numbers: numbers n whose divisors can be partitioned into two disjoint sets whose sums are both sigma(n)/2.
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21
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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; internal format)
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OFFSET
| 1,1
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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. [From T. D. Noe (noe(AT)sspectra.com), Mar 31 2010]
A179529(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 19 2010]
A118372 is a subsequence, the S-perfect numbers. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 28 2010]
All 205283 odd abundant numbers less than 10^8 that have even abundance are Zumkeller numbers.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers [From Peter Luschny (peter(AT)luschny.de), Dec 14 2009]
Peter Luschny, Zumkeller Numbers. [From Peter Luschny (peter(AT)luschny.de), Dec 14 2009]
Reinhard Zumkeller, Illustration of initial terms
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EXAMPLE
| n=48: 1+3+4+6+8+16+24=2+12+48, therefore 48 is a term (A083206(48)=5).
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MAPLE
| Contribution from Peter Luschny (peter(AT)luschny.de), Dec 14 2009: (Start)
with(numtheory); with(combstruct);
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(NULL) 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; if Found then n else NULL fi end: (End)
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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] [From T. D. Noe (noe(AT)sspectra.com), Mar 31 2010]
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CROSSREFS
| Cf. A083209, A083211, A000203, A005101, A000396, A005835.
Cf. A048055, A171641 [From Peter Luschny (peter(AT)luschny.de), Dec 14 2009]
Sequence in context: A087883 A162416 A105455 * A145278 A094371 A189793
Adjacent sequences: A083204 A083205 A083206 * A083208 A083209 A083210
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KEYWORD
| nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2003
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EXTENSIONS
| Name improved by T. D. Noe (noe(AT)sspectra.com), Mar 31 2010
Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010
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